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1 International Mathematics Olympiad 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1st IMO, Romania, 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2nd IMO, Romania, 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3rd IMO, Hungary, 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4th IMO, Czechoslovakia, 1962 . . . . . . . . . . . . . . . . . . . . . . . . . 5th IMO, Poland, 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6th IMO, USSR, 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7th IMO, West Germany, 1965 . . . . . . . . . . . . . . . . . . . . . . . . . 8th IMO, Bulgaria, 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9th IMO, Yugoslavia, 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 7 8 9 10 11 11 12 13 14 16 16 18

1.10 10th IMO, USSR, 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 11th IMO, Romania, 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 12th IMO, Hungary, 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 13th IMO, Czechoslovakia, 1971 . . . . . . . . . . . . . . . . . . . . . . . . 1

2

CONTENTS 1.14 14th IMO, USSR, 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 15th IMO, USSR, 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 16th IMO, West Germany, 1974 . . . . . . . . . . . . . . . . . . . . . . . . 1.17 17th IMO, Bulgaria, 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 18th IMO, Austria, 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.19 19th IMO, Yugoslavia, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.20 20th IMO, Romania, 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.21 21st IMO, United Kingdom, 1979 . . . . . . . . . . . . . . . . . . . . . . . 1.22 22nd IMO, Washington, USA, 1981 . . . . . . . . . . . . . . . . . . . . . . 1.23 23rd IMO, Budapest, Hungary, 1982 . . . . . . . . . . . . . . . . . . . . . . 1.24 24th IMO, Paris, France, 1983 . . . . . . . . . . . . . . . . . . . . . . . . . 1.25 25th IMO, Prague, Czechoslovakia, 1984 . . . . . . . . . . . . . . . . . . . 1.26 26th IMO, Helsinki, Finland, 1985 . . . . . . . . . . . . . . . . . . . . . . . 1.27 27th IMO, Warsaw, Poland, 1986 . . . . . . . . . . . . . . . . . . . . . . . 1.28 28th IMO, Havana, Cuba , 1987 . . . . . . . . . . . . . . . . . . . . . . . . 1.29 29th IMO, Camberra, Australia, 1988 . . . . . . . . . . . . . . . . . . . . . 1.30 30th IMO, Braunschweig, West Germany, 1989 . . . . . . . . . . . . . . . . 1.31 31st IMO, Beijing, People’s Republic of China, 1990 . . . . . . . . . . . . . 1.32 32nd IMO, Sigtuna, Sweden, 1991 . . . . . . . . . . . . . . . . . . . . . . . 1.33 33rd IMO, Moscow, Russia, 1992 . . . . . . . . . . . . . . . . . . . . . . . . 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 36 37 38

CONTENTS 1.34 34th IMO, Istambul, Turkey, 1993 . . . . . . . . . . . . . . . . . . . . . . . 1.35 35th IMO, Hong Kong, 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . 1.36 36th IMO, Toronto, Canada, 1995 . . . . . . . . . . . . . . . . . . . . . . . 1.37 37th IMO, Mumbai, India, 1996 . . . . . . . . . . . . . . . . . . . . . . . . 1.38 38th IMO, Mar del Plata, Argentina, 1997 . . . . . . . . . . . . . . . . . . 1.39 39th IMO, Taipei, Taiwan, 1998 . . . . . . . . . . . . . . . . . . . . . . . . 1.40 40th IMO, Bucharest, Romania, 1999 . . . . . . . . . . . . . . . . . . . . . 1.41 41st IMO, Taejon, South Korea, 2000 . . . . . . . . . . . . . . . . . . . . . 1.42 42nd IMO, Washington DC, USA, 2001 . . . . . . . . . . . . . . . . . . . . 1.43 43rd IMO, Glascow, United Kingdom, 2002 . . . . . . . . . . . . . . . . . . 1.44 44th IMO, Tokyo, Japan, 2003 . . . . . . . . . . . . . . . . . . . . . . . . .

3 39 40 41 42 43 44 45 46 47 48 49

2 William Lowell Putnam Competition 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 46th Anual William Lowell Putnam Competition, 1985 . . . . . . . . . . . 47th Anual William Lowell Putnam Competition, 1986 . . . . . . . . . . . 48th Anual William Lowell Putnam Competition, 1987 . . . . . . . . . . . 49th Anual William Lowell Putnam Competition, 1988 . . . . . . . . . . . 50th Anual William Lowell Putnam Competition, 1989 . . . . . . . . . . . 51th Anual William Lowell Putnam Competition, 1990 . . . . . . . . . . . 52th Anual William Lowell Putnam Competition, 1991 . . . . . . . . . . . 53th Anual William Lowell Putnam Competition, 1992 . . . . . . . . . . .

50 50 52 54 56 58 60 62 64

4 2.9

CONTENTS 54th Anual William Lowell Putnam Competition, 1993 . . . . . . . . . . . 66 68 69 71 73 75 76 79 80 81

2.10 55th Anual William Lowell Putnam Competition, 1994 . . . . . . . . . . . 2.11 56th Anual William Lowell Putnam Competition, 1995 . . . . . . . . . . . 2.12 57th Anual William Lowell Putnam Competition, 1996 . . . . . . . . . . . 2.13 58th Anual William Lowell Putnam Competition, 1997 . . . . . . . . . . . 2.14 59th Anual William Lowell Putnam Competition, 1998 . . . . . . . . . . . 2.15 60th Anual William Lowell Putnam Competition, 1999 . . . . . . . . . . . 2.16 61st Anual William Lowell Putnam Competition, 2000 . . . . . . . . . . . . 2.17 62nd Anual William Lowell Putnam Competition, 2001 . . . . . . . . . . . 2.18 63rd Anual William Lowell Putnam Competition, 2002 . . . . . . . . . . .

3 Asiatic Paci?c Mathematical Olympiads 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 1st Asiatic Paci?c Mathematical Olympiad, 1989 . . . . . . . . . . . . . . . 2nd Asiatic Paci?c Mathematical Olympiad, 1990 . . . . . . . . . . . . . . 3rd Asiatic Paci?c Mathematical Olympiad, 1991 . . . . . . . . . . . . . . 4th Asiatic Paci?c Mathematical Olympiad, 1992 . . . . . . . . . . . . . . . 5th Asiatic Paci?c Mathematical Olympiad, 1993 . . . . . . . . . . . . . . . 6th Asiatic Paci?c Mathematical Olympiad, 1994 . . . . . . . . . . . . . . . 7th Asiatic Paci?c Mathematical Olympiad, 1995 . . . . . . . . . . . . . . . 8th Asiatic Paci?c Mathematical Olympiad, 1996 . . . . . . . . . . . . . . . 9th Asiatic Paci?c Mathematical Olympiad, 1997 . . . . . . . . . . . . . . .

84 84 85 86 86 87 88 89 90 91

CONTENTS 3.10 10th Asiatic Paci?c Mathematical Olympiad, 1998 . . . . . . . . . . . . . . 3.11 11th Asiatic Paci?c Mathematical Olympiad, 1999 . . . . . . . . . . . . . . 3.12 12th Asiatic Paci?c Mathematical Olympiad, 2000 . . . . . . . . . . . . . . 3.13 13th Asiatic Paci?c Mathematical Olympiad, 2001 . . . . . . . . . . . . . . 3.14 14th Asiatic Paci?c Mathematical Olympiad, 2002 . . . . . . . . . . . . . . 3.15 15th Asiatic Paci?c Mathematical Olympiad, 2003 . . . . . . . . . . . . . .

5 92 93 93 94 95 96

Chapter 1 International Mathematics Olympiad

1.1 1st IMO, Romania, 1959

21n + 4 14n + 3 is irreductible for every natural number n

1. Prove that the fraction

2. For what real values of x is x+ √ 2x ? 1 + x? √ 2x ? 1 = A

√ given (a) A = 2, (b) A = 1, (c) A = 2, where only non-negative real numbers are admitted for square roots? 3. Let a, b, c be real numbers. Consider the quadratic equation in cos x: a cos 2 x + b cos x + c = 0. Using the numbers a, b and c, form a quadratic ecuation in cos 2x, whose roots are the same as those of the original ecuation. Compare the ecuations in cos x and cos 2x for a = 4, b = 2 and c = ?1 4. Construct a right triangle with hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. 5. An arbitrary point M is selected in the interior of the segment AB . The squares AM CD and M BEF are constructed on the same side of AB , sith the segments AM and M B as their respective bases. The circles circumscribed abut these squares, 6

1.2. 2N D IMO, ROMANIA, 1960

7

with centers P and Q intersect at M and also at another point N . Let N denote the intersection of the straight lines AF and BC . (a) Prove that the points N and N coinside. (b) Prove that the straight lines M N pass throught a ?xed point S independent of the choice of M . (c) Find the locus of the midpoints of the the segment P Q as M varies between A and B . 6. Two planes, P and Q, intersect along the line p. The point A is given in the plane P , and the point C in the plane Q; neither of these points lies on the straight line p. Construct an isosceles trapezoid ABCD (with AB parallel to CD ) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.

1.2

2nd IMO, Romania, 1960

1. Determine all three-digit numbers N having the property that N is divisible by 11, N and 11 is equal to the sum of the squares of the digits of N . 2. For what values of the variable x does the following inequality hold? 4x2 √ 1 ? 1 + 2x

2

< 2x + 9

3. In a given right triangle ABC , the hypotenuse BC , of lenght a, is dividen into n equal parts (n an odd integer). Let α be the acute angle subtending, from A, that segment which contains the middle point of the hypotenuse. Let h be the lenght of the altitude to the hypotenuse of the triangle. Prove: tan α = (n2 4nh ? 1) a

4. Construct a triangle ABC , given ha , hb (the altitudes fron A and B ) and ma , the median from vertex A. 5. Consider the cube ABCDA B C D (whith face ABCD directly above face A B C D ). (a) Find the locus of the midpoints of segment XY , where X is any point of AC and Y is any point of B D .

8

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD (b) Find the locus of points Z which lie on the segment XY of part (a) with ZY = 2XZ . 6. Considere a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let V1 be the volume of the cone and V2 the volumen of the cilinder. (a) Prove that V1 = V2 . (b) Find the smallest number k for which V1 = kV2 , for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone. 7. An isosceles trapezoid with bases a and c, and altitude h is given. (a) On the axis of symmetry of this trapezoid, ?nd all points P such that both legs of the trapezoid subtended right angles at P . (b) Calculate the distance of P from either base. (c) Determine under what conditions such points P actually exist. (Discuss varius case that might arise)

1.3

3rd IMO, Hungary, 1961

1. Solve the system of equations: x+y+z = a x + y 2 + z 2 = b2 xy = z 2

2

where a and b are constants. Give the conditions that a and b must satisfy so that x, y, z (the solutions of the system) are distinct positive numbers. √ 2. Let a, b, c the sides of a triangle, and T its area. Prove: a2 + b2 + c2 ≥ 4 3T . In what case does the equality hold? 3. Solve the equation cosn x ? sinn x = 1, where n is a natural number. 4. Consider the triangle P1 P2 P3 and a point P within the triangle. Lines P P1 , P P2 , P P3 intersect the opposite side in points Q1 , Q2 , Q3 respectively. Prove that, of the P1 P P2 P P3 P numbers P , P , P at least one is less than or equal to 2 and at least one is Q1 Q2 Q3 grater than or equal to 2.

1.4. 4T H IMO, CZECHOSLOVAKIA, 1962

9

5. Construct triangle ABC if AC = b, AB = c and AM B = ω , where M is the midpoint of the segment BC and ω < 90? . Prove that a solution exists and only if b tan ω ≥ c < b. In what case does the equality hold? 2 6. Considere a plane ε and three non-collinear points A, B, C on the same side of ε; suppose the plane determined by these three points is not parallel to ε. In plane a take three arbitrary points A , B , C . Let L, M, N be the midpoints of segments AA , BB , CC ; let G the centroid of triangle LM N (We will not considere positions of A , B , C such that the points L, M , N do not form a triangle) What is the locus of point G as A , B , C range independently over the plane ε?

1.4

4th IMO, Czechoslovakia, 1962

1. Find the smallest natural number n which has the following properties: (a) Its decimal representation has 6 as the last digit. (b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number n 2. Determine all real number x which satisfy the inequality: √ √ 1 3?x? x+1> 2 3. Consider the cube ABCDA B C D (ABCD and A B C D are the upper and lower bases, respectively, and edges AA , BB , CC , DD are parallel) The point X moves at constant speed along the perimeterof the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B C CB in the direction B C CBB . Points X and Y begin their motion at the same instant from the starting position A and B , respectively. Determine and draw the locus of the midpoints of the segment XY . 4. Solve the ecuation cos2 x + cos2 2x + cos2 3x = 1 5. On the circle K there are given three distinct points A, B , C . Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the cuadrilateral thus obtained.

10

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. Considere an isosceles triangle. let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is d = r (r ? 2ρ) 7. The tetrahedon SABC has the following propoerty: there exists ?ve spheres, each tangent to the edges SA, SB, SC, BC, CA, AB or their extentions. (a) Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron ?ve such spheres exist.

1.5

5th IMO, Poland, 1963

√ √ 2 x ? p + 2 x2 ? 1 = x, where p is a real param-

1. Find all real roots of the equation eter.

2. Point A and segment BC are given. Determine the locus of points in space which are vertices of right angles with one side passing throught A, and the other side intersecting the segment BC . 3. In an n?gon all of whose interior angles are equal, the lenght of consecutive sides satisfy the relation a1 ≥ a2 ≥ · ≥ an . Prove that a1 = a2 = · = an . 4. Find all solution x1 , x2 , x3 , x4 , x5 of the system (1) (2) (3) (4) (5) where y is a parameter 5. Prove that cos 2π 3π 1 π ? cos + cos = 7 7 7 2 x 5 + x2 x 1 + x3 x 2 + x4 x 3 + x5 x 4 + x1 = = = = = yx1 yx2 yx3 yx4 yx5

6. Five students, A, B, C, D, E , took part in a contest. One prediction was that contestants would ?nish in the order ABCDE . This prediction was very poor. In fact no contestant ?nished in the position predicted, and no two contestants predicted to ?nish consecutively actually did so. A second prediction has the contestants ?nishing

1.6. 6T H IMO, USSR, 1964

11

in the order DAECB . This prediction was better. Exactly two of the contestants ?nished in the places predicted, and two disjoint pairs of students predicted to ?nish consecutively actually did so. Determine the order in which the contestants ?nished.

1.6

6th IMO, USSR, 1964

(b) Prove that there is not positive integer n such that 2n + 1 is dibisible by 7.

1. (a) Find all positive integers n for which 2n ? 1 is divisible by 7. 2. Let a, b, c be the sides of a triangle. Prove that

a2 (b + c ? a) + b2 (c + a ? b) + c2 (a + b ? c) ≤ 3abc 3. A circle is inscribed in triangle ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts o? a triangle from ABC . In each of these triangle, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c) 4. Seventeen people correspond by mail with one another, each one with all the rest. In their letters only three di?erent topics are discussed. Each pair of correspondent deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic. 5. Suppose ?ve points in a plane are situated so that no two of the straight lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have. 6. In tetrahedron ABCD , vertex D is connected with D0 the centroid of ABC . Lines parallel to DD0 are drawn through A, B and C . These lines intersect the planes BCD, CAD and ABD in points A1 , B1 and C1 , respectively. Prove that the volume of ABCD is one third the volume of A1 B1 C1 D0 . Is the result true if point D0 is selected anywhere within ABC ?

1.7

7th IMO, West Germany, 1965

1. Determine all value x in the interval 0 ≤ x ≤ 2π which satisfy the inequality √ √ √ 2 cos x ≤ 1 + sin 2x ? 1 ? sin 2x ≤ 2

12

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Consider the system of equations a11 x1 + a12 x2 + a13 x3 = 0 a21 x1 + a22 x2 + a23 x3 = 0 a31 x1 + a32 x2 + a33 x3 = 0 with unknowns x1 , x2 , x3 . The coe?cient satisfy the conditions: (a) a11 , a22 , a33 are positive numbers; (b) the remaining coe?cients are negative numbers; (c) in each equation, the sum of the coe?cient is positive . Prove that the given system has only the solution x1 = x2 = x3 = 0. 3. Given the tetrahedron ABCD whose edges AB and CD have lenght a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω . Tetrahedron ABCD is divided into two solid by plane ε, parallel to lines AB and CD . The ratio of the distances of ε from AB and CD is equal to k . Compute the ratio of the volumes of the two solids obtained. 4. Find all sets of four real numbers x1 , x2 , x3 , x4 such that the sum of any one and the product of the other three is equal to 2. 5. Consider OAB with acute angle AOB . Through a point M = O perpendiculars are dawn to OA and OB , the feet of which are P and Q respectively. The point of intersection of the altitudes of OP Q is H . What is the locus of H if M is permitted to range over (a) the side AB ? (b) the interior of OAB ?

6. In a plane a set of n points (n ≥ 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segment. We de?ne a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.

1.8

8th IMO, Bulgaria, 1966

1. In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the

1.9. 9T H IMO, YUGOSLAVIA, 1967

13

contestants who did not solve problem A, the number who solved B was twice the number who solved C . The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. How many students solved only problem B ? 2. Let a, b, c be the lengths of the sides of a triangle and α, β, γ , respectively, the (a tan α + b tan β ), the triangle angles opposite these sides. Prove tat if a + b = tan γ 2 is isosceles. 3. Prove: The sum of the distances of the vertices of a regular tetrahedron from the centre of its circumscribed sphere is less than the sum of the distances of these vertices from any other poin in space. 4. Prove that for every natural number n, and for every real number x = non-negative integer and k any integer), 1 1 1 + +···+ = cot x ? cot 2n x sin 2x sin 4x sin 2n x 5. Solve the system of equations |a1 ? a2 |x2 + |a1 ? a3 |x3 + |a1 ? a4 |x4 |a2 ? a1 |x2 + |a2 ? a3 |x3 + |a2 ? a4 |x4 |a3 ? a1 |x1 + |a3 ? a2 |x2 + |a3 ? a4 |x4 |a4 ? a1 |x1 + |a4 ? a2 |x2 + |a4 ? a3 |x3 where a1 , a2 , a3 , a4 are four di?erent real numbers. 6. In the interior of sides BC, CA, AB of triangle ABC , any points K, L, M , respectively, are selected. Prove that the area of at least one of the triangle AM L, BKM , is less than or equal to one quarter of the area of ABC CLK = = = = 1 1 1 1

kπ 2t

(t any

1.9

9th IMO, Yugoslavia, 1967

1. Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with BAD = α. If ABD is acute, prove that the four circles of√ radius 1 with centers A, B, C, D cover the parallelogram if and only if a ≤ cos α + 3 sin α. 2. Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume 1 is smaller than or equal to 8

14

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 3. Let k, m, n be natural numbers such that m + k + 1 is a prime greater than n + 1. Let cs = s (s + 1). Prove that the product (cm+1 ? ck ) (cm+2 ? ck ) · · · (cm+n ? ck ) is divisible by the product c1 c2 · · · cn . 4. Let A0 B0 C0 and A1 B1 C1 be any two acute-angled triangles. Consider all triangles ABC that are similar to A1 B1 C1 and circumscribed about triangle A0 B0 C0 (where A0 lies on BC , B0 on CA and C0 on AB ) Of all such triangles, determine the one with maximum area, and construct it. 5. Consider the sequence {cn }, where c1 = a 1 + a 2 + · · · + a 8 2 2 c2 = a 2 1 + a2 + · · · + a 8 . . .

n n cn = a n 1 + a2 + · · · + a 8 . . .

in which a1 , a2 , . . . , a8 are real numbers not all equal to zero. Suppose that an in?nite number of terms of the sequence {cn } are equal to zero. Find all natural numbers for which cn = 0. 6. In a sport contest, there were m medals awarded on n successive days (n > 1). On 1 of the remaining medals were awarded. On the second the ?rst day, one medal and 7 1 of the now remaining medals were awarded; and so on. On day, two medals and 7 the n-th and last day, the remaining n medals were awarded. How many days did the contest last. and how many medals were awarded altogether?

1.10

10th IMO, USSR, 1968

1. Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another. 2. Find all natural numbers x such that the product of their digits (in decimal notation) is equal to x2 ? 10x ? 22.

1.10. 10T H IMO, USSR, 1968 3. Consider the system of equations: ax2 1 ax2 2 + + bx1 bx2 + c = x2 + c = x3 . . .

15

ax2 n?1 + bxn?1 + c = xn ax2 + bxn + c = x1 n with unknowns x1 , x2 , . . . , xn , where a, b, c are real and a = 0. Let 4ac. Prove that for this system (a) If (b) If (c) If < 0, ther is no solution, = 0, ther is exactly one solution, > 0, ther is more than one solution. = (b ? 1)2 ?

4. Prove than in every tetrahedon there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle. 5. Let f be a real-valued function de?ned for all real numbers x such that, for some positive constant a, the equation f (x + a) = holds for all x (a) Prove that the function f is periodic (i.e. there exists a positive number b such that f (x + b) = f (x) for all x) (b) For a = 1, give an example of a non-constant function with the requiered properties. 6. For every natural number n, evaluate the sum

∞ k =0

1 + 2

f (x) ? [f (x)]2

n+1 n+2 n + 2k n + 2k = +··· + + · · · + 2k+1 2 4 2k+1

(the symbol x denotes the greatest integer not exceding x).

16

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

1.11

11th IMO, Romania, 1969

1. Prove that there are in?nitely many numbers a with the following property: the number z = n4 + a is not prime for any natural number n. 2. Let a1 , a2 , . . . , an be real variable, and f (x) = cos (a1 + x) + 1 1 1 cos (a2 + x) + cos (a3 + x) + · · · + n?1 cos (an + x) 2 4 2

Given that f (x1 ) = f (x2 ) = 0, prove that x2 ? x1 = mπ for some integer m. 3. For each value of k = 1, 2, 3, 4, 5, ?nd necessary and su?cient conditions on the number a > 0 so that there exist a tetrahedron with k edges of length a, and the remaining 6 ? k edges of lenght 1. 4. A semicircular arc γ is drawn on AB as diameter. C is a point on γ other than A and B , and D is the foot of the perpendicular from C to AB . We consider three circles γ1 , γ2 , γ3 , all tangent to the line AB . Of these, γ1 is inscrived in ABC , while γ2 and γ3 are both tangent to CD and to γ , one on each side of CD . Prove that γ1 , γ2 and γ3 have a second tangent in common. 5. Given n > 4 points in the plane such that no three are collinear. Prove that there 3 are at least n? convex quadrilaterals whose vertices are four of the given points. 2

2 6. Prove that for all real numbers x1 , x2 , y1 , y2 , z1 , z2 with x1 > 0, x2 > 0, x1 y1 ? z1 > 2 0, x2 y2 ? z2 > 0, the inequality

1 8 1 + 2 ≤ 2 2 x1 y 1 ? z 1 x2 y 2 ? z 2 (x1 + x2 ) (y1 + y2 ) ? (z1 + z2 ) is satis?ed. Give necessary and su?cient conditions for equality.

1.12

12th IMO, Hungary, 1970

1. Let M be a point on the sede AB of ABC . Let r1 , r2 and r be the radii of the inscribed circles of the triangles AM C , BM C and ABC . Let q1 , q2 and q be the radii of the excribed circles of the same triangles that lie in the angle ACB . Prove that r r1 r2 · = q1 q2 q

1.12. 12T H IMO, HUNGARY, 1970

17

2. Let a, b and n be integers greater than 1, and let a and b be the two bases of two number systems. An?1 and An are numbers in the system with base a and Bn?1 and Bn are numbers in the system with base b; these are related as follows: An = xn xn?1 · · · x0 Bn = xn xn?1 · · · x0 such that xn = 0 and xn?1 = 01 . Prove that Bn?1 An?1 < ? ?a > b An Bn 3. The real numbers a0 , a1 , . . . , an , . . . satisfy the condition 1 = a0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ ·. The numbers b1 , b2 , . . . , bn , . . . are de?ned by

n

An?1 = xn?1 xn?2 · · · x0 Bn?1 = xn?1 xn?2 · · · x0

bn =

k =1

1?

ak?1 ak

1 √ ak

(a) Prove that 0 ≤ bn < 2 for all n. (a) Given c with 0 ≤ c < 2, prove that there exist numbers a0 , a1 , . . . such that bn > c for large enough n. 4. Find the set of all positive integers n with the property that the set {n, n + 1, n + 2, n + 3, n + 4, n + 5} can be partitioned into sets such that the product of the numbers in one set equals the product of the numbers in the other set 5. In the tetrahedron ABCD , the angle BDC is a right angle. Suppose that the foot H of the perpendicular from D to the plane ABC is the intersection of the altitudes of ABC . Prove that (AB + BC + CA)2 ≤ 6 AD 2 + BD 2 + CD 2 For what tetrahedra does equality hold? 6. In the plane are 100 points, no three of them are collinear. Consider all posible triangles having these points as vertices. Prove that no more than 70% of these triangles are acute-angled.

1

The xi ’s are the digits in the respective bases, and of course, all of them are lower than the lowest base

18

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

1.13

13th IMO, Czechoslovakia, 1971

1. Prove that the following assertion is true for n = 3 and n = 5, and that it is false for every other natural number n > 2. If a1 , a2 , . . . , an are arbitrary real numbers, then: (a1 ? a2 ) (a1 ? a3 ) · · · (a1 ? an ) + (a2 ? a1 ) (a2 ? a3 ) · · · (a2 ? an ) + · · · + (an ? a1 ) (an ? a2 ) · · · (an ? an?1 ) ≤ 0 2. Consider a convex polyhedron P1 with nine vertices A1 A2 , ..., A9 ; let Pi be the polyhedron obtained from P1 by a translation that moves vertex A1 to Ai (i = 2, 3, ..., 9). Provethat at least two of the polyhedra P1 , P2 , ..., P9 have an interiorpoint in common. 3. Prove that the set of integers of the form 2k ? 3(k = 2, 3, ...) contains an in?nite subset in which every two members are relatively prime. 4. All the faces of tetrahedron ABCD are acute-angled triangles. We consider all closed polygonal paths of the form XY ZT X de?ned as follows: X is a point on edge AB distinct from A and B ; similarly, Y, Z, T are interior points of edges BCCD, DA, respectively. Prove: (a) If DAB + BCD = CDA + ABC, then among thepolygonal paths, there is none of minimal length. (b) If DAB + BCD = CDA + ABC, then there arein?nitely many shortest polygonal paths, their common length being 2AC sin(α/2), where α = BAC + CAD + DAB. 5. Prove that for every natural number m, there exists a ?nite set S of points in a plane with the following property: For every point A in S, there are exactly m points in S which are at unit distance from A. 6. Let A = (aij )(i, j = 1, 2, ..., n) be a square matrix whose elements are non-negative integers. Suppose that whenever an element aij = 0, the sum of the elements in the ith row and the j th column is ≥ n. Prove that the sum of all the elements of the matrix is ≥ n2 /2.

1.14

14th IMO, USSR, 1972

1. Prove that from a set of ten distinct two-digit numbers (in the decimalsystem), it is possible to select two disjoint subsets whose members havethe same sum.

1.15. 15T H IMO, USSR, 1973

19

2. Prove that if n ≥ 4, every quadrilateral that can be inscribed in acircle can be dissected into n quadrilaterals each of which is inscribablein a circle. 3. Let m and n be arbitrary non-negative integers. Prove that (2m)!(2n)! m!n!(m + n)! is an integer. (0! = 1) 4. Find all solutions (x1 , x2 , x3 , x4 , x5 ) of the system of inequalities

2 (x2 1 ? x3 x5 )(x2 ? x3 x5 ) 2 (x2 2 ? x4 x1 )(x3 ? x4 x1 ) 2 (x2 3 ? x5 x2 )(x4 ? x5 x2 ) 2 (x2 4 ? x1 x3 )(x5 ? x1 x3 ) 2 (x2 5 ? x2 x4 )(x1 ? x2 x4 )

≤ ≤ ≤ ≤ ≤

0 0 0 0 0

where x1 , x2 , x3 , x4 , x5 are positive real numbers. 5. Let f and g be real-valued functions de?ned for all real values of xand y, and satisfying the equation f (x + y ) + f (x ? y ) = 2f (x)g (y ) for all x, y. Prove that if f (x) is not identically zero, and if |f (x)| ≤ 1 for all x, then |g (y )| ≤ 1 for all y. 6. Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

1.15

15th IMO, USSR, 1973

? ? → ? ? → ? ? → 1. Point O lies on line g ; OP1 , OP2 , . . . , OPn are unit vectors such that points P1 , P2 , ..., Pn all lie in a plane containing g and on one side of g. Prove that if n is odd, ? ? → ? ? → ? ? → OP1 + OP2 + · · · + OPn ≥ 1 ? ? → ? ? → Here OM denotes the length of vector OM .

20

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Determine whether or not there exists a ?nite set M of points in spacenot lying in the same plane such that, for any two points A and B of M,one can select two other points C and D of M so that lines AB and CD are parallel and not coincident. 3. Let a and b be real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. For all such pairs (a, b), ?nd the minimum value of a2 + b 2 . 4. A soldier needs to check on the presence of mines in a region having theshape of an equilateral triangle. The radius of action of his detector isequal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission? 5. G is a set of non-constant functions of the real variable x of the form f (x) = ax + b, a and b are real numbers, and G has the following properties: (b) If f is in G, then its inverse f ?1 is in G; here the inverse of f (x) = ax + b is f ?1 (x) = (x ? b)/a. (c) For every f in G, there exists a real number xf such that f (xf ) = xf . Prove that there exists a real number k such that f (k ) = k for all f in G. 6. Let a1 , a2 , ..., an be n positive numbers, and let q be a givenreal number such that 0 < q < 1. Find n numbers b1 , b2 , ..., bn forwhich (b) q < (a) ak < bk for k = 1, 2, · · · , n,

bk+1 bk

(a) If f and g are in G, then g ? f is in G; here (g ? f )(x) = g [f (x)].

<

1 q

for k = 1, 2, ..., n ? 1,

1+q (a1 1?q

(c) b1 + b2 + · · · + bn <

+ a2 + · · · + an ).

1.16

16th IMO, West Germany, 1974

1. Three players A, B and C play the following game: On each of three cardsan integer is written. These three numbers p, q, r satisfy 0 < p < q < r. Thethree cards are shu?ed and one is dealt to each player. Each then receivesthe number of counters indicated by the card he holds. Then the cards areshu?ed again; the counters remain with the players.

1.17. 17T H IMO, BULGARIA, 1975

21

This process (shu?ing, dealing, giving out counters) takes place for at least two rounds. After the last round, A has 20 counters in all, B has 10 and C has 9. At the last round B received r counters. Who received q counters on the ?rst round? 2. In the triangle ABC prove that there is a point D on side AB suchthat CD is the geometric mean of AD and DB if and only if sin A sin B ≤ sin2 3. Prove that the number

n k =0 2n+1 2k +1

C . 2

23k is not divisible by 5 for any integer n ≥ 0.

4. Consider decompositions of an 8 × 8 chessboard into p non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If ai is the number of white squares in the i-th rectangle, then a1 < a2 < · · · < ap . Find the maximum value of p for which such a decomposition is possible. For this value of p, determine all possible sequences a1 , a2 , · · · , ap . 5. Determine all possible values of S= a b c d + + + a+b+d a+b+c b+c+d a+c+d

where a, b, c, d are arbitrary positive numbers. 6. Let P be a non-constant polynomial with integer coe?cients. If n(P ) isthe number of distinct integers k such that (P (k ))2 = 1, prove that n(P ) ? deg(P ) ≤ 2, where deg(P ) denotes the degree of the polynomial P.

1.17

17th IMO, Bulgaria, 1975

x1 ≥ x2 ≥ · · · ≥ xn and y1 ≥ y2 ≥ · · · ≥ yn

1. Let xi , yi (i = 1, 2, ..., n) be real numbers such that

Prove that, if z1 , z2 , · · · , zn is any permutation of y1 , y2 , · · · , yn , then

n i=1 n

(xi ? yi )2 ≤

i=1

(xi ? zi )2

22

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Let a1 , a2 , a3, · · · be an in?nite increasing sequence of positive integers. Prove that for every p ≥ 1 there are in?nitely many am which can be written in the form am = xap + yaq with x, y positive integers and q > p. 3. On the sides of an arbitrary triangle ABC, triangles ABR, BCP, CAQ areconstructed externally with CBP = CAQ = 45? , BCP = ACQ = 30? , ABR = BAR = 15? . Prove that QRP = 90? and QR = RP. 4. When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A and B are written in decimal notation.) 5. Determine, with proof, whether or not one can ?nd 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number. 6. Find all polynomials P, in two variables, with the following properties: (i) for a positive integer n and all real t, x, y P (tx, ty ) = tn P (x, y ) (that is, P is homogeneous of degree n), (ii) for all real a, b, c, P (b + c, a) + P (c + a, b) + P (a + b, c) = 0 (iii) P (1, 0) = 1.

1.18

18th IMO, Austria, 1976

1. In a plane convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths ofthe other diagonal. 2. Let P1 (x) = x2 ? 2 and Pj (x) = P1 (Pj ?1 (x)) for j = 2, 3, · · ·.Show that, for any positive integer n, the roots of the equation Pn (x) = x are real and distinct.

1.19. 19T H IMO, YUGOSLAVIA, 1977

23

3. A rectangular box can be ?lled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can ?ll exactly 40% ofthe box. Determine the possible dimensions of all such boxes. 4. Determine, with proof, the largest number which is the product of positiveintegers whose sum is 1976. 5. Consider the system of p equations in q = 2p unknowns x1 , x2 , · · · , xq : a11 x1 + a12 x2 + · · · + a1q xq = 0 a21 x1 + a22 x2 + · · · + a2q xq = 0 ··· ap1 x1 + ap2 x2 + · · · + apq xq = 0 with every coe?cient aij member of the set {?1, 0, 1}. Prove that the system has a solution (x1 , x2 , · · · , xq ) such that (b) there is at least one value of j for which xj = 0, (c) |xj | ≤ q (j = 1, 2, ..., q ). 6. A sequence {un } is de?ned by u0 = 2, u1 = 5/2, un+1 = un (u2 n?1 ? 2) ? u1 for n = 1, 2, · · · Prove that for positive integers n, [un ] = 2[2

n ?(?1)n ]/3

(a) all xj (j = 1, 2, ..., q ) are integers,

where [x] denotes the greatest integer ≤ x.

1.19

19th IMO, Yugoslavia, 1977

1. Equilateral triangles ABK, BCL, CDM, DAN are constructed inside the square ABCD. Prove that the midpoints of the four segments KL, LM, M N, N K and the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN are the twelve vertices of a regular dodecagon.

24

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. In a ?nite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive.Determine the maximum number of terms in the sequence. 3. Let n be a given integer > 2, and let Vn be the set of integers 1+ kn, where k = 1, 2, .... A number m ∈ Vn is called indecomposable in Vn if there do not exist numbers p, q ∈ Vn such that pq = m. Prove that there exists a number r ∈ Vn that can be expressed as the product of elements indecomposable in Vn in more than one way. (Products which di?er only in the order of their factors will be considered the same.) 4. Four real constants a, b, A, B are given, and f (θ ) = 1 ? a cos θ ? b sin θ ? A cos 2θ ? B sin 2θ Prove that if f (θ ) ≥ 0 for all real θ , then a2 + b2 ≤ 2 and A2 + B 2 ≤ 1. 5. Let a and b be positive integers. When a2 + b2 is divided by a + b,the quotient is q and the remainder is r. Find all pairs (a, b) suchthat q 2 + r = 1977. 6. Let f (n) be a function de?ned on the set of all positive integers and having all its values in the same set. Prove that if f (n + 1) > f (f (n)) for each positive integer n, then f (n) = n for each n.

1.20

20th IMO, Romania, 1978

1. m and n are natural numbers with 1 ≤ m < n. In their decimal representations, the last three digits of 1978m are equal, respectively, to the last three digits of 1978n . Find m and n such that m + n has its least value. 2. P is a given point inside a given sphere. Three mutually perpendicular rays from P intersect the sphere at points U, V, and W ; Q denotes the vertex diagonally opposite to P in the parallelepiped determined by P U, P V, and P W. Find the locus of Q for all such triads of rays from P . 3. The set of all positive integers is the union of two disjoint subsets {f (1), f (2), ..., f (n), ...}, {g (1), g (2), ..., g (n), ...}, where f (1) < f (2) < · · · < f (n) < · · ·

1.21. 21ST IMO, UNITED KINGDOM, 1979 g (1) < g (2) < · · · < g (n) < · · · and g (n) = f (f (n)) + 1 for all n ≥ 1 Determine f (240).

25

4. In triangle ABC, AB = AC. A circle is tangent internally to thecircumcircle of triangle ABC and also to sides AB, AC at P, Q, respectively. Prove that the midpoint of segment P Q is the center of the incircle of triangle ABC. 5. Let {ak }(k = 1, 2, 3, ..., n, ...) be a sequence of distinct positive integers. Prove that for all natural numbers n, n n 1 ak ≥ 2 k =1 k k =1 k 6. An international society has its members from six di?erent countries. The list of members contains 1978 names, numbered 1, 2, ..., 1978. Prove that there is at least one member whose number is the sum of thenumbers of two members from his own country, or twice as large as the numberof one member from his own country.

1.21

21st IMO, United Kingdom, 1979

p 1 1 1 1 1 = 1? + ? +···? + q 2 3 4 1318 1319

1. Let p and q be natural numbers such that

Prove that p is divisible by 1979. 2. A prism with pentagons A1 A2 A3 A4 A5 and B1 B2 B3 B4 B5 as top and bottom faces is given. Each side of the two pentagons and each of the line-segments Ai Bj for all i, j = 1, ..., 5, is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a di?erent color. Show that all 10 sides of the top and bottom faces are the same color. 3. Two circles in a plane intersect. Let A be one of the points of intersection. Starting simultaneously from A two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a ?xed point P in the plane such that, at any time, the distances from P to the moving points are equal.

26

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Given a plane π, a point P in this plane and a point Q not in π, ?nd all points R in π such that the ratio (QP + P A)/QR is a maximum. 5. Find all real numbers a for which there exist non-negative real numbers x1 , x2 , x3 , x4 , x5 satisfying the relations

5 5 5

kxk = a,

k =1 k =1

k 3 xk = a 2 ,

k =1

k 5 xk = a 3

6. Let A and E be opposite vertices of a regular octagon. A frog starts jumping at vertex A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches vertex E, the frog stops and stays there.. Let a n be the number of distinct paths of exactly n jumps ending at E. Prove that a2n?1 = 0, 1 a2n = √ (xn?1 ? y n?1 ), n = 1, 2, 3, · · · , 2 √ √ where x = 2 + 2 and y = 2 ? 2

Note. A path of n jumps is a sequence of vertices (P0 , ..., Pn ) such that (i) P0 = A, Pn = E ; (iii) for every i, 0 ≤ i ≤ n ? 1, Pi and Pi+1 are adjacent. (ii) for every i, 0 ≤ i ≤ n ? 1, Pi is distinct from E ;

1.22

22nd IMO, Washington, USA, 1981

1. P is a point inside a given triangle ABC.D, E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P for which BC CA AB + + PD PE PF is least. 2. Let 1 ≤ r ≤ n and consider all subsets of r elements of theset {1, 2, ..., n}. Each of these subsets has a smallest member. Let F (n, r ) denote the arithmetic mean of these smallest numbers; prove that F (n, r ) = n+1 r+1

1.23. 23RD IMO, BUDAPEST, HUNGARY, 1982

27

3. Determine the maximum value of m3 + n3 ,where m and n are integers satisfying m, n ∈ {1, 2, ..., 1981} and (n2 ? mn ? m2 )2 = 1 4. (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n ? 1 numbers? (b) For which values of n > 2 is there exactly one set having the stated property? 5. Three congruent circles have a common point O and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point O are collinear. 6. The function f (x, y ) satis?es (1) f (0, y ) = y + 1, (2)f (x + 1, 0) = f (x, 1), (3) f (x + 1, y + 1) = f (x, f (x + 1, y )), for all non-negative integers x, y. Determine f (4, 1981).

1.23

23rd IMO, Budapest, Hungary, 1982

1. The function f (n) is de?ned for all positive integers n and takes on non-negative integer values. Also, for all m, n f (m + n) ? f (m) ? f (n) = 0 or 1 f (2) = 0, f (3) > 0, and f (9999) = 3333 Determine f (1982). 2. A non-isosceles triangle A1 A2 A3 is given with sides a1 , a2 , a3 (ai is the side opposite Ai ). For all i = 1, 2, 3, Mi is the midpoint of side ai , and Ti . is the pointwhere the incircle touches side ai . Denote by Si the re?ection of Ti in the interior bisector of angle Ai . Prove that the lines M1 , S1 , M2 S2 , and M3 S3 are concurrent. 3. Consider the in?nite sequences {xn } of positive real numbers with the following properties: x0 = 1, and for all i ≥ 0, xi+1 ≤ xi

28

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD (a) Prove that for every such sequence, there is an n ≥ 1 such that x2 x2 x2 0 + 1 + · · · + n?1 ≥ 3.999 x1 x2 xn (b) Find such a sequence for which x2 x2 x2 0 + 1 + · · · + n?1 < 4 x1 x2 xn 4. Prove that if n is a positive integer such that the equation x3 ? 3xy 2 + y 3 = n has a solution in integers (x, y ), then it has at least three suchsolutions. Show that the equation has no solutions in integers when n = 2891 5. The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N , respectively, so that CN AM = =r AC CE Determine r if B, M, and N are collinear. 6. Let S be a square with sides of length 100, and let L be a path within S which does not meet itself and which is composed of line segments A0 A1 , A1 A2 , · · · , An?1 An with A0 = An . Suppose that for every point P of the boundary of S there is a point of L at a distance from P not greater than 1/2. Prove that there are two points X and Y in L such that the distance between X and Y is not greater than 1, and the length of that part of L which lies between X and Y is not smaller than 198.

1.24

24th IMO, Paris, France, 1983

1. Find all functions f de?ned on the set of positive real numbers which take positive real values and satisfy the conditions: (i) f (xf (y )) = yf (x) for all positive x, y ; (ii) f (x) → 0 as x → ∞ 2. Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1 and O2 , respectively. One of the common tangents to the circles touches C1 at P1 and C2 at P2 , while the other touches C1 at Q1 and C2 at Q2 . Let M1 be the midpoint of P1 Q1 ,and M2 be the midpoint of P2 Q2 . Prove that O1 AO2 = M1 AM2 .

1.25. 25T H IMO, PRAGUE, CZECHOSLOVAKIA, 1984

29

3. Let a, b and c be positive integers, no two of which have a common divisor greater than 1. Show that 2abc ? ab ? bc ? ca is the largest integer which cannot be expressed in the form xbc + yca + zab,where x, y and z are non-negative integers. 4. Let ABC be an equilateral triangle and E the set of all points contained in the three segments AB, BC and CA (including A, B and C ). Determine whether, for every partition of E into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer. 5. Is it possible to choose 1983 distinct positive integers, all less than or equal to 10 5 , no three of which are consecutive terms of an arithmetic progression? Justify your answer. 6. Let a, b and c be the lengths of the sides of a triangle. Prove that a2 b(a ? b) + b2 c(b ? c) + c2 a(c ? a) ≥ 0 Determine when equality occurs.

1.25

25th IMO, Prague, Czechoslovakia, 1984

0 ≤ yz + zx + xy ? 2xyz ≤

1. Prove that

7 27 where x, y and z arenon-negative real numbers for which x + y + z = 1.

2. Find one pair of positive integers a and b such that: (i) ab(a + b) is not divisible by 7; (ii) (a + b)7 ? a7 ? b7 is divisible by 77 . Justify your answer. 3. In the plane two di?erent points O and A are given. For each point X of the plane, other than O , denote by a(X ) the measure of the angle between OA and OX in radians, counterclockwise from OA(0 ≤ a(X ) < 2π ). Let C (X ) be the circle with center O and radius of length OX + a(X )/OX. Each point of the plane is colored by one of a ?nite number ofcolors. Prove that there exists a point Y for which a(Y ) > 0 such that its color appears on the circumference of the circle C (Y ).

30

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Let ABCD be a convex quadrilateral such that the line CD is a tangent to the circle on AB as diameter. Prove that the line AB is a tangent to the circle on CD as diameter if and only if the lines BC and AD are parallel. 5. Let d be the sum of the lengths of all the diagonals of a plane convex polygon with n vertices (n > 3), and let p be its perimeter. Prove that n?3< n 2d < p 2 n+1 ?2 2

where [x] denotes the greatest integer not exceeding x. 6. Let a, b, c and d be odd integers such that 0 < a < b < c < d and ad = bc. Prove that if a + d = 2k and b + c = 2m for some integers k and m, then a = 1.

1.26

26th IMO, Helsinki, Finland, 1985

1. A circle has center on the side AB of the cyclic quadrilateral ABCD . The other three sides are tangent to the circle. Prove that AD + BC = AB 2. Let n and k be given relatively prime natural numbers, k < n. Each number in the set M = {1, 2, ..., n ? 1} is colored either blue or white. It is given that (ii) for each i ∈ M, i = k both i and |i ? k | have the same color. Prove that all numbers in M must have the same color. 3. For any polynomial P (x) = a0 + a1 x + · · · + ak xk with integer coe?cients, the number of coe?cients which are odd is denoted by w (P ). For i = 0, 1, ..., let Qi (x) = (1 + x)i . Prove that if i1 i2 , ..., in are integers such that 0 ≤ i1 < i2 < · · · < in then w (Qi1 + Qi2 + · · · + Qin ) ≥ w (Qi1 ) 4. Given a set M of 1985 distinct positive integers, none of which has a prime divisor greater than 26. Prove that M contains at least one subset of four distinct elements whose product is the fourth power of an integer. 5. A circle with center O passes through the vertices A and C of triangle ABC and intersects the segments AB and BC again at distinct points K and N, respectively. The circumscribed circles of the triangles ABC and EBN intersect at exactly two distinct points B and M. Provethat angle OM B is a right angle. (i) for each i ∈ M, both i and n ? i have the same color;

1.27. 27T H IMO, WARSAW, POLAND, 1986 6. For every real number x1 , construct the sequence x1 , x2 , ... by setting xn+1 = xn xn + 1 for each n ≥ 1 n

31

Prove that there exists exactly one value of x1 for which 0 < xn < xn+1 < 1 for every n.

1.27

27th IMO, Warsaw, Poland, 1986

1. Let d be any positive integer not equal to 2, 5, or 13. Show that one canind distinct a, b in the set {2, 5, 13, d} such that ab ? 1 is not perfect square. 2. A triangle A1 A2 A3 and a point P0 are given in the plane. We de?ne As = As?3 for all s ≥ 4. We construct a set of points P1 , P2 , P3 , . . . , such that Pk+1 is the image of Pk under a rotation withenter Ak+1 through angle 120? clockwise (for k = 0, 1, 2,ldots). Prove that if P1986 = P0 , then the triangle A1 A2 A3 equilateral. 3. To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all ?ve numbers is positive. If three consecutive vertices are assigned the numbers x, y , z respectively and y < 0 then the following operation is allowed: the numbers x, y , z are replaced by x + y , ?y , z + y respectively. Such an operation is performed repeatedly as long as at least one of the ?ve numbers is negative. Determine whether this procedure necessarily comes to and end after a ?nite number of steps. 4. Let A, B be adjacent vertices of a regular n-gon (n ≥ 5) in thelane having center at O . A triangle XY Z , which is congruent to andnitially conincides with OAB , moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X . 5. Find all functions f , de?ned on the non-negative real numbers and taking nonnegative real values, such that: (i) f (xf (y ))f (y ) = f (x + y ) for all x, y ≥ 0,

(ii) f (2) = 0,

(iii) f (x) = 0 for 0 ≤ x < 2.

32

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. One is given a ?nite set of points in the plane, each point having integeroordinates. Is it always possible to color some of the points in the set rednd the remaining points white in such a way that for any straight line Larallel to either one of the coordinate axes the di?erence (in absolutealue) between the numbers of white point and red points on L is not greaterhan 1?

1.28

28th IMO, Havana, Cuba , 1987

1. Let pn (k ) be the number of permutations of the set {1, . . . , n}, n ≥ 1, which have exactly k ?xed points. Prove that

n k =0

k · pn (k ) = n!

(Remark: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S is called a ?xed point of the permutation f if f (i) = i.) 2. In an acute-angled triangle ABC the interior bisector of the angle A intersects BC at L and intersects the circumcircle of ABC again at N . From point L perpendiculars are drawn to AB and AC , the feet of theseerpendiculars being K and M respectively. Prove that the quadrilateral AKN M and the triangle ABC have equal areas.

2 2 3. Let x1 , x2 , . . . , xn be real numbers satisfying x2 1 + x2 + · · · + xn = 1. Prove that for every integer k ≥ 2 there are integers a1 , a2 , . . . , an , not all 0, such that |ai | ≤ k ? 1 For all i and √ (k ? 1) n |a1 x1 + a1 x2 + · · · + an xn | ≤ kn ?

4. Prove that there is no function f from the set of non-negative integers into itself such that f (f (n)) = n + 1987 for every n. 5. Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. 6. Let n be an integer greater than or equal to 2. Prove that if k 2 + k + n is prime for all integers k such that 0 ≤ k ≤ n , then k 2 + k + n is prime for all integers k such 3 that 0 ≤ k ≤ n ? 2.

1.29. 29T H IMO, CAMBERRA, AUSTRALIA, 1988

33

1.29

29th IMO, Camberra, Australia, 1988

1. Consider two coplanar circles of radii R and r (R > r ) with the same center. Let P be a ?xed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C . The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A = P .) (i) Find the set of values of BC 2 + CA2 + AB 2 . (ii) Find the locus of the midpoint of BC . 2. Let n be a positive integer and let A1 , A2 , . . . , A2n+1 be subsets of a set B . Suppose that (a) Each Ai has exactly 2n elements, (b) Each Ai ∩ Aj (1 ≤ i < j ≤ 2n + 1) contains exactly one element, and (c) Every element of B belongs to at least two of the Ai . For which values of n can one assign to every element of B one of the numbers 0 and 1 in such a way that Ai has 0 assigned to exactly n of its elements? 3. A function f is de?ned on the positive integers by f (1) f (2n) f (4n + 1) f (4n + 3) for all positive integers n. Determine the number of positive integers n, less than or equal to 1988, for which f (n) = n. 4. Show that set of real numbers x which satisfy the inequality k 5 ≥ 4 k =1 x ? k is a union of disjoint intervals, the sum of whose lengths is 1988. 5. ABC is a triangle right-angled at A, and D is the foot of the altituderom A. The straight line joining the incenters of the triangles ABD , ACD intersects the sides AB , AC at the points K , L respectively. S and T denote the areas of the triangles ABC and AKL respectively.how that S ≥ 2T .

70

= = = =

1, f (3) = 3, f (n), 2f (2n + 1) ? f (n), 3f (2n + 1) ? 2f (n),

34

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. Let a and b be positive integers such that ab + 1 divides a2 + b2 . Show that a2 + b 2 ab + 1 is the square of an integer.

1.30

30th IMO, Braunschweig, West Germany, 1989

1. Prove that the set {1, 2, . . . , 1989} can be expressed as the disjoint union of subsets Ai (i = 1, 2, . . . , 117) such that: (i) Each Ai contains 17 elements; (ii) The sum of all the elements in each Ai is the same. 2. In an acute-angled triangle ABC the internal bisector of angle A meets their cumcircle of the triangle again at A1 . Points B1 and C1 are de?ned similarly. Let A0 be the point of intersection of the line AA1 with the external bisectors of angles B and C . Points B0 and C0 are de?ned similarly. Prove that: (i) The area of the triangle A0 B0 C0 is twice the area of the hexagon AC1 BA1 CB1 . (ii) The area of the triangle A0 B0 C0 is at least four times the area of the triangle ABC . 3. Let n and k be positive integers and let S be a set of n points in the plane such that (i) No three points of S are collinear, and (ii) For any point P of S there are at least k points of S equidistant from P . Prove that: k< 1 √ + 2n 2

4. Let ABCD be a convex quadrilateral such that the sides AB , AD , BC satisfy AB = AD + BC . There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC . Show that: 1 1 1 √ ≥√ +√ h AD BC 5. Prove that for each positive integer n there exist n consecutive positive integers none of which is an integral power of a prime number.

1.30. 30T H IMO, BRAUNSCHWEIG, WEST GERMANY, 1989

35

6. A permutation (x1 , x2 , . . . , xm ) of the set {1, 2, . . . , 2n}, where n is a positive integer, is said to have property P if |xi ? xi+1 | = n for at least one i in {1, 2, . . . , 2n ? 1}. Show that, for each n, there are more permutations with property P than without.

36

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD

1.31

31st IMO, Beijing, People’s Republic of China, 1990

1. Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB . The tangent line at E to the circle through D , E , and M intersects the lines BC and AC at F and G, respectively. If AM = t, AB ?nd in terms of t. 2. Let n ≥ 3 and consider a set E of 2n ? 1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E . Find the smallest value of k so that every such coloring of k points of E is good. 3. Determine all integers n > 1 such that 2n + 1 n2 is an integer. 4. Let Q+ be the set of positive rational numbers. Construct a function f : Q+ → Q+ such that f (x) f (xf (y )) = y + for all x, y in Q . 5. Given an initial integer n0 > 1, two players, A and B , choose integers n1 , n2 , n3 , . . . alternately according to the following rules: Knowing n2k , A chooses any integer n2k+1 such that n2k ≤ n2k+1 ≤ n2 2k . Knowing n2k+1 , B chooses any integer n2k+2 such that n2k+1 n2k+2 EG EF

1.32. 32N D IMO, SIGTUNA, SWEDEN, 1991 is a prime raised to a positive integer power.

37

Player A wins the game by choosing the number 1990; player B wins by choosing the number 1. For which n0 does: (b) B have a winning strategy? (a) A have a winning strategy? (c) Neither player have a winning strategy?

6. Prove that there exists a convex 1990-gon with the following two properties: (a) All angles are equal. (b) The lengths of the 1990 sides are the numbers 12 , 22 , 32 , . . . , 19902 in some order.

1.32

32nd IMO, Sigtuna, Sweden, 1991

2. Let n > 6 be an integer and a1 , a2 , . . . , ak be all the natural numbers less than n and relatively prime to n. If a2 ? a1 = a3 ? a2 = · · · = ak ? ak?1 > 0 prove that n must be either a prime number or a power of 2. 3. Let S = {1, 2, 3, . . . , 280}. Find the smallest integer n such that each n-element subset of S contains ?ve numbers which are pairwise relatively prime. 4. Suppose G is a connected graph with k edges. Prove that it is possible to label the edges 1, 2, . . . , k in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [A graph consists of a set of points, called vertices , together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices u, v belongs to at most one edge. The graph G is connected if for each pair of distinct vertices x, y there is some sequence of vertices x = v0 , v1 , v2 , . . . , vm = y such that each pair vi , vi+1 (0 ≤ i < m) is joined by an edge of G.]

1. Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the angles A, B, C meet the opposite sides in A , B , C respectively. Prove that AI · BI · CI 8 1 < ≤ 4 AA · BB · CC 27

38

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 5. Let ABC be a triangle and P an interior point of ABC . Show that at least one of the angles P AB, P BC, P CA is less than or equal to 30? . 6. An in?nite sequence x0 , x1 , x2 , . . . of real numbers is said to be bounded if there is a constant C such that |xi | ≤ C for every i ≥ 0.

Given any real number a > 1, construct a bounded in?nite sequence x0 , x1 , x2 , . . . such that |xi ? xj ||i ? j |a ≥ 1 for every pair of distinct nonnegative integers i, j .

1.33

33rd IMO, Moscow, Russia, 1992

(a ? 1)(b ? 1)(c ? 1) is a divisor of abc ? 1

1. Find all integers a, b, c with 1 < a < b < c such that

2. Let R denote the set of all real numbers. Find all functions f : R → R such that f x2 + f (y ) = y + (f (x))2 forall x, y ∈ R

3. Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of n such that whenever exactly n edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color. 4. In the plane let C be a circle, L a line tangent to the circle C, and M a point on L. Find the locus of all points P with the following property: there exists two points Q, R on L such that M is the midpoint of QR and C is the inscribed circle of triangle P QR. 5. Let S be a ?nite set of points in three-dimensional space. Let Sx , Sy , Sz be the sets consisting of the orthogonal projections of the points of S onto the yz -plane, zx-plane, xy -plane, respectively. Prove that |S |2 ≤ |Sx | · |Sy | · |Sz | where |A| denotes the number of elements in the ?nite set |A|. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)

1.34. 34T H IMO, ISTAMBUL, TURKEY, 1993

39

6. For each positive integer n, S (n) is de?ned to be the greatest integer such that, for every positive integer k ≤ S (n), n2 can be written as the sum of k positive squares. (a) Prove that S (n) ≤ n2 ? 14 for each n ≥ 4.

(b) Find an integer n such that S (n) = n2 ? 14.

(c) Prove that there are in?ntely many integers n such that S (n) = n2 ? 14.

1.34

34th IMO, Istambul, Turkey, 1993

1. Let f (x) = xn + 5xn?1 + 3, where n > 1 is an integer. Prove that f (x) cannot be expressed as the product of two nonconstant polynomials with integer coe?cients. 2. Let D be a point inside acute triangle ABC such that ∠ADB = AC · BD = AD · BC . (a) Calculate the ratio (AB · CD )/(AC · BD ). (b) Prove that the tangents at C to the circumcircles of perpendicular. ACD and BCD are ACB + π/2 and

3. On an in?nite chessboard, a game is played as follows. At the start, n2 pieces are arranged on the chessboard in an n by n block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of n for which the game can end with only one piece remaining on the board. 4. For three points P, Q, R in the plane, we de?ne m(P QR) as the minimum length of the three altitudes of P QR. (If the points are collinear, we set m(P QR) = 0.) Prove that for points A, B, C, X in the plane, m(ABC ) ≤ m(ABX ) + m(AXC ) + m(XBC ) 5. Does there exist a function f : N → N such that f (1) = 2, f (f (n)) = f (n) + n for all n ∈ N, and f (n) < f (n + 1) for all n ∈ N?

40

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. There are n lamps L0 , . . . , Ln?1 in a circle (n > 1), where we denote Ln+k = Lk . (A lamp at all times is either on or o?.) Perform steps s0 , s1 , . . . as follows: at step si , if Li?1 is lit, switch Li from on to o? or vice versa, otherwise do nothing. Initially all lamps are on. Show that: (a) There is a positive integer M (n) such that after M (n) steps all the lamps are on again; (b) If n = 2k , we can take M (n) = n2 ? 1; (c) If n = 2k + 1, we can take M (n) = n2 ? n + 1.

1.35

35th IMO, Hong Kong, 1994

1. Let m and n be positive integers. Let a1 , a2 , . . . , am be distinct elements of {1, 2, . . . , n} such that whenever ai + aj ≤ n for some i, j , 1 ≤ i ≤ j ≤ m, there exists k , 1 ≤ k ≤ m, with ai + aj = ak . Prove that a1 + a 2 + · · · + a m n+1 ≥ m 2 2. ABC is an isosceles triangle with AB = AC . Suppose that (a) M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB ; (b) Q is an arbitrary point on the segment BC di?erent from B and C ; (c) E lies on the line AB and F lies on the line AC such that E , Q, F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF . 3. For any positive integer k , let f (k ) be the number of elements in the set {k + 1, k + 2, . . . , 2k } whose base 2 representation has precisely three 1s. (a) Prove that, for each positive integer m, there exists at least onepositive integer k such that f (k ) = m. (b) Determine all positive integers m for which there exists exactly one k with f (k ) = m.

1.36. 36T H IMO, TORONTO, CANADA, 1995 4. Determine all ordered pairs (m, n) of positive integers such that n3 + 1 mn ? 1 is an integer.

41

5. Let S be the set of real numbers strictly greater than ?1. Find all functions f : S → S satisfying the two conditions: (a) f (x + f (y ) + xf (y )) = y + f (x) + yf (x) for all x and y in S ; (b)

f (x) x

is strictly increasing on each of the intervals ?1 < x < 0 and 0 < x.

6. Show that there exists a set A of positive integers with the following property: For any in?nite set S of primes there exist two positive integers m ∈ A and n ∈ / A each of which is a product of k distinct elements of S for some k ≥ 2.

1.36

36th IMO, Toronto, Canada, 1995

1. Let A, B, C, D be four distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . The line XY meets BC at Z . Let P be a point on the line XY other than Z . The line CP intersects the circle with diameter AC at C and M , and the line BP intersects the circle with diameter BD at B and N . Prove that the lines AM, DN, XY are concurrent. 2. Let a, b, c be positive real numbers such that abc = 1. Prove that a3 (b 1 1 1 3 + 3 + 3 ≥ + c) b (c + a) c (a + b) 2

3. Determine all integers n > 3 for which there exist n points A1 , . . . , An in the plane, no three collinear, and real numbers r1 , . . . , rn such that for 1 ≤ i < j < k ≤ n, the area of Ai Aj Ak is ri + rj + rk . 4. Find the maximum value of x0 for which there exists a sequence x0 , x1 . . . , x1995 of positive reals with x0 = x1995 , such that for i = 1, . . . , 1995, xi?1 + 2 xi?1 = 2xi + 1 xi

42

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 5. Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF = F A, such that BCD = EF A = π/3. Suppose G and H are points in theinterior of the hexagon such that AGB = DHE = 2π/3. Provethat AG + GB + GH + DH + HE ≥ CF . 6. Let p be an odd prime number. How many p-element subsets A of {1, 2, . . . 2p} are there, the sum of whose elements is divisible by p?

1.37

37th IMO, Mumbai, India, 1996

1. We are given a positive integer r and a rectangular board ABCD with dimensions |AB | = 20, |BC | = 12. The rectangle is divided into a grid of 20 × 12 unit squares. The following moves are permitted on the board: one can move from √one square to another only if the distance between the centers of the two squares is r . The task is to ?nd a sequence of moves leading from the square with A as a vertex to the square with B as a vertex. (a) Show that the task cannot be done if r is divisible by 2 or 3. (b) Prove that the task is possible when r = 73. (c) Can the task be done when r = 97? 2. Let P be a point inside triangle ABC such that AP C ? ABC

AP B ? ACB =

Let D, E be the incenters of triangles AP B, AP C , respectively. Show that AP, BD, CE meet at a point. 3. Let S denote the set of nonnegative integers. Find all functions f from S to itself such that f (m + f (n)) = f (f (m)) + f (n) ?m, n ∈ S 4. The positive integers a and b are such that the numbers 15a + 16b and 16a ? 15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? 5. Let ABCDEF be a convex hexagon such that AB is parallel to DE , BC is parallel to EF , and CD is parallel to F A. Let RA , RC , RE denote the circumradii of triangles F AB, BCD, DEF , respectively, and let P denote the perimeter of the hexagon.

1.38. 38T H IMO, MAR DEL PLATA, ARGENTINA, 1997 Prove that RA + R C + R E ≥

43

P 2

6. Let p, q, n be three positive integers with p + q < n. Let (x0 , x1 , . . . , xn ) be an (n + 1)tuple of integers satisfying the following conditions: (a) x0 = xn = 0. (b) For each i with 1 ≤ i ≤ n, either xi ? xi?1 = p or xi ? xi?1 = ?q . Show that there exist indices i < j with (i, j ) = (0, n), such that xi = xj .

1.38

38th IMO, Mar del Plata, Argentina, 1997

1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard). For any pair of positive integers m and n, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths m and n,ie along edges of the squares. Let S1 be the total area of the black part of the triangle and S2 be the total area of thehite part. Let f (m, n) = |S1 ? S2 | (a) Calculate f (m, n) for all positive integers m and n which are eitheroth even or both odd. (b) Prove that f (m, n) ≤

1 2

(c) Show that there is no constant C such that f (m, n) < C for all m and n. endenumerate

max{m, n} for all m and n.

2. The angle at A is the smallest angle of triangle ABC . The points B and C divide the circumcircle of the triangle into two arcs. Let U be an interior point of the arc between B and C which does not contain A. The perpendicular bisectors of AB and AC meet the line AU at V and W , respectively. The lines BV and CW meet at T . Show that AU = T B + T C 3. Let x1 , x2 , . . . , xn be real numbers satisfying the conditions |x1 + x2 + · · · + xn | = 1

44 and

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD n+1 i = 1, 2, . . . , n 2 Show that there exists a permutation y1 , y2 , . . . , yn of x1 , x2 , . . . , xn such that |xi | ≤ |y1 + 2y2 + · · · + nyn | ≤ n+1 2

4. An n × n matrix whose entries come from the set S = {1, 2, . . . , 2n ? 1} is called a silver matrix if, for each i = 1, 2, . . . , n, the ith row and the ith column together contain all elements of S . Show that (a) there is no silver matrix for n = 1997; (b) silver matrices exist for in?nitely many values of n. 5. Find all pairs (a, b) of integers a, b ≥ 1 that satisfy the equation ab = b a 6. For each positive integer n , let f (n) denote the number of ways of representing n as a sum of powers of 2 with nonnegative integer exponents.epresentations which di?er only in the ordering of their summands are considered to be the same. For instance, f (4) = 4, because the number 4 can be represented in the following four ways: 4; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1. Prove that, for any integer n ≥ 3, 2n

2 /4 2

< f (2n ) < 2n

2 /2

.

1.39

39th IMO, Taipei, Taiwan, 1998

1. In the convex quadrilateral ABCD , the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P , where the perpendicular bisectors of AB and DC meet, is inside ABCD . Prove that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP have equal areas. 2. In a competition, there are a contestants and b judges, where b ≥ 3 is an odd integer. Each judge rates each contestant as either pass or fail. Suppose k is a number such that, for any two judges, their ratings coincide for at most k contestants. Prove that k/a ≥ (b ? 1)/(2b).

1.40. 40T H IMO, BUCHAREST, ROMANIA, 1999

45

3. For any positive integer n, let d(n) denote the number of positive divisors of n (including 1 and n itself). Determine all positive integers k such that d(n2 )/d(n) = k for some n. 4. Determine all pairs (a, b) of positive integers such that ab2 + b + 7 divides a2 b + a + b. 5. Let I be the incenter of triangle ABC . Let the incircle of ABC touch the sides BC , CA, and AB at K , L, and M , respectively. The line through B parallel to M K meets the lines LM and LK at R and S , respectively. Prove that angle RIS is acute. 6. Consider all functions f from the set N of all positive integers into itself satisfying f (t2 f (s)) = s(f (t))2 for all s and t in N . Determine the least possible value of f (1998).

1.40

40th IMO, Bucharest, Romania, 1999

1. Determine all ?nite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S , the perpendicular bisector of the line segment AB is an axis of symmetry for S . 2. Let n be a ?xed integer, with n ≥ 2. (a) Determine the least constant C such that the inequality

2 ? xi xj (x2 i + xj ) ≤ C

1≤i<j ≤n

?

1≤i≤n

holds for all real numbers x1 , · · · , xn ≥ 0.

xi ?

?4

(b) For this constant C , determine when equality holds. 3. Consider an n × n square board, where n is a ?xed even positive integer. The board is divided into n2 unit squares. We say that two di?erent squares on the board are adjacent if they have a common side. N unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least one marked square. Determine the smallest possible value of N .

46

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Determine all pairs (n, p) of positive integers such that p is a prime, n not exceeded 2p, and (p ? 1)n + 1 is divisible by np?1 . 5. Two circles G1 and G2 are contained inside the circle G, and are tangent to G at the distinct points M and N , respectively. G1 passes through the center of G2 . The line passing through the two points of intersection of G1 and G2 meets G at A and B . The lines M A and M B meet G1 at C and D , respectively. Prove that CD is tangent to G2 . 6. Determine all functions f : R ?→ R such that f (x ? f (y )) = f (f (y )) + xf (y ) + f (x) ? 1 for all real numbers x, y .

1.41

41st IMO, Taejon, South Korea, 2000

1. Two circles ω1 and ω2 intersect at M and N . Line is tangent to the circles at A and B , respectively, so that M lies closer to than N . Line CD , with C on ω1 and D on ω2 , is parallel to and passes through M . Let lines AC and BD meet at E ; let lines AN and CD meet at P ; and let lines BN and CD meet at Q. Prove that EP = EQ. 2. Let a, b, c be positive real numbers such that abc = 1. Prove that: a?1+ 1 b b?1+ 1 c c?1+ 1 ≤1 a

3. Let n ≥ 2 be a positive integer. Initially, there are n ?eas on a horizontal line, not all at the same point. For a positive real number λ, de?ne a move as follows: choose any two ?eas at points A and B , with A to the left of B ; let the ?ea at A jump to the point C on the line to the right of B with BC = λ. AB Determine all the values of λ such that for any point M on the line and any initial position of the n ?eas, there is a sequence of moves that will take all the ?eas to the position to the right of M .

1.42. 42N D IMO, WASHINGTON DC, USA, 2001

47

4. A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contain at least one card. A member of the audience selects two of the three boxes, choose one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identi?es the box from which no card has been choosen. How many ways are there to put all the cards into the boxes so that this trick always works? (Two ways are considered diferent if at least one of card is put nto a di?erent box) 5. Determine whether or not there exists a positive integer n such that: n is divisible by exactely 2000 di?erent prime numbers, and 2n + 1 is divisible by n. 6. Let AH1 , BH2 , CH3 be the altitudes of an acute-angled triangle ABC . The incircle of the triangle ABC touches the sides BC , CA, AB at T1 , T2 , T3 , respectively. Let the lines 1 , 2 , 3 be the re?ections of the lines H2 H3 , H3 H1 H1 H2 in the lines T2 T3 , T3 T1 , T1 T2 , respectively. Prove that 1 , 2 , triangle ABC .

3

determine a triangle whose vertices lie on the incircle of the

1.42

42nd IMO, Washington DC, USA, 2001

1. Let ABC be an acute-angled triangle with circumcentre O . Let P on BC be the foot of the altitude from A. Suppose that BCA ≥ ABC + 30? . Prove that CAB + COP < 90? 2. Prove that a b c +√ 2 +√ 2 ≥1 + 8bc b + 8ca c + 8ab for all positive real numbers a, b and c. √ a2

3. Twenty-one girls and twenty-one boys took part in a mathematical contest. ? Each contestant solved at most six problems.

? For each girl and each boy, at least one problem was solved by both of them Prove that there was a problem that was solved by at least three girls and at least three boys.

48

CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Let n be an odd integer greater than 1, and let k1 , k2 , . . . , kn be given integers. For each of the n! permutations a = (a1 , a2 , . . . , an ) of 1, 2, . . . , n, let

n

S (a) =

i=1

ki ai

Prove that there are two permutations b and c, such that n! is a divisor of S (b) ? S (c). 5. In a triangle ABC , let AP bisect BAC , with P on BC , and let BQ bisect ABC , with Q on CA. It is known that BAC = 60? and that AB + BP = AQ + QB . What are the possible angles of triangle ABC ? 6. Let a, b, c, d be integers with a > b > c > d > 0. Suppose that ac + bd = (b + d + a ? c) (b + d ? a + c) Prove ab + cd is not a prime.

1.43

43rd IMO, Glascow, United Kingdom, 2002

1. S is the set of all (h, k ) with h, k non-negative integers such that h + k < n. Each element of S is colored red or blue, so that if (h, k ) is red and h ≤ h, k ≤ k , then (h , k ) is also red. A type 1 subset of S has n blue elements with di?erent ?rst member and a type 2 subset of S has n blue elements with di?erent second member. Show that there are the same number of type 1 and type 2 subsets. 2. BC is a diameter of a circle center O . A is any point on the circle with ∠AOC > 60o . EF is the chord which is the perpendicular bisector of AO. D is the midpoint of the minor arc AB . The line through O parallel to AD meets AC at J . Show that J is the incenter of triangle CEF . 3. Find all pairs of integers m > 2, n > 2 such that there are in?nitely many positive integers k for which k n + k 2 ? 1 divides k m + k ? 1. 4. The positive divisors of the integer n > 1 are d1 < d2 < . . . < dk , so that d1 = 1, dk = n. Let d = d1 d2 + d2 d3 + · · · + dk?1 dk . (a) Prove that D < n2 . (b) Determine all n for which D is a divisor of n2 .

1.44. 44T H IMO, TOKYO, JAPAN, 2003 5. Find all functions f from the set R of real numbers to itself such that (f (x) + f (z ))(f (y ) + f (t)) = f (xy ? zt) + f (xt + yz ) for all x, y, z, t in R.

49

6. n > 2 circles of radius 1 are drawn in the plane so that no line meets more than two 1)π of the circles. Their centers are O1 , O2 , · · · , On . Show that 1≤i<j ≤n Oi1 ≤ (n? Oj 4

1.44

44th IMO, Tokyo, Japan, 2003

1. Let A be a 101-element subset of the set S = {1, 2, . . . , 1000000}. Prove that there exist numbers t1 , t2 , . . . , t100 in S such that the sets Aj = {x + tj | x ∈ A} j = 1, 2, . . . , 100 are pairwise disjoint. 2. Find all pairs (m, n) of positive integers such that

m2 2mn2 ?n3 +1

is a positive integer.

3. A convex hexagon is given in which any two √ opposite sides have the following property: the distance between their midpoints is 3/2 times the sum of their lengths. Prove that all the angles of the hexagon are equal. (A convex ABCDEF has three pairs of opposite sides: AB and DE , BC and EF , CD and F A.) 4. Let ABCD be a cyclic quadrilateral. Let P, Q and R be the feet of perpendiculars from D to lines BC, CA and AB , respectively. Showhat P Q = QR if and only if the bisectors of angles ABC and ADC meet on segment AC . 5. Let n be a positive integer and x1 , x2 , . . . , xn be real numbers with x1 ≤ x2 ≤ . . . ≤ xn . (a) Prove that

? ?

n n

i=1 j =1

(b) Show that the equality holds if and only if x1 , x2 , . . . , xn form an arithmetic sequence. 6. Show that for each prime p, there exists a prime q such that np ? p is not divisible by q for any positive integer n.

|xi ? xj |? ≤

?2

2(n2 ? 1) n n (xi ? xj )2 3 i=1 j =1

Chapter 2 William Lowell Putnam Competition

2.1 46th Anual William Lowell Putnam Competition, 1985

1. Determine, with proof, the number of ordered triples (A1 , A2 , A3 ) of sets which have the property that (i) A1 ∪ A2 ∪ A3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and

(ii) A1 ∩ A2 ∩ A3 = ?.

Express your answer in the form 2a 3b 5c 7d , where a, b, c, d are nonnegative integers. 2. Let T be an acute triangle. Inscribe a rectangle R in T with one side along a side of T . Then inscribe a rectangle S in the triangle formed by the side of R opposite the side on the boundary of T , and the other two sides of T , with one side along the side of R. For any polygon X , let A(X ) denote the area of X . Find the maximum value, )+A(S ) or show that no maximum exists, of A(R , where T ranges over all triangles and A(T ) R, S over all rectangles as above. 3. Let d be a real number. For each integer m ≥ 0, de?ne a sequence {am (j )}, j = 0, 1, 2, . . . by the condition am (0) = d/2m , am (j + 1) = (am (j ))2 + 2am (j ), Evaluate limn→∞ an (n). 50 j ≥ 0.

2.1. 46T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1985

51

4. De?ne a sequence {ai } by a1 = 3 and ai+1 = 3ai for i ≥ 1. Which integers between 00 and 99 inclusive occur as the last two digits in the decimal expansion of in?nitely many ai ? 5. Let Im = Im = 0?

2π 0

cos(x) cos(2x) · · · cos(mx) dx. For which integers m, 1 ≤ m ≤ 10 is

6. If p(x) = a0 + a1 x + · · · + am xm is a polynomial with real coe?cients ai , then set

2 2 Γ(p(x)) = a2 0 + a1 + · · · + a m .

Let F (x) = 3x2 + 7x + 2. Find, with proof, a polynomial g (x) with real coe?cients such that (i) g (0) = 1, and (ii) Γ(f (x)n ) = Γ(g (x)n ) for every integer n ≥ 1. 7. Let k be the smallest positive integer for which there exist distinct integers m1 , m2 , m3 , m4 , m5 such that the polynomial p(x) = (x ? m1 )(x ? m2 )(x ? m3 )(x ? m4 )(x ? m5 ) has exactly k nonzero coe?cients. Find, with proof, a set of integers m1 , m2 , m3 , m4 , m5 for which this minimum k is achieved. 8. De?ne polynomials fn (x) for n ≥ 0 by f0 (x) = 1, fn (0) = 0 for n ≥ 1, and d fn+1 (x) = (n + 1)fn (x + 1) dx for n ≥ 0. Find, with proof, the explicit factorization of f100 (1) into powers of distinct primes. 9. Let a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 . . . . . . . . . ... ... ... .. .

be a doubly in?nite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that am,n > mn for some pair of positive integers (m, n).

52

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. Let C be the unit circle x2 +y 2 = 1. A point p is chosen randomly on the circumference C and another point q is chosen randomly from the interior of C (these points are chosen independently and uniformly over their domains). Let R be the rectangle with sides parallel to the x and y -axes with diagonal pq . What is the probability that no point of R lies outside of C ? √ ?1 2 ∞ 11. Evaluate 0∞ t?1/2 e?1985(t+t ) dt. You may assume that ?∞ e?x dx = π . 12. Let G be a ?nite set of real n × n matrices {Mi }, 1 ≤ i ≤ r , which form a group under matrix multiplication. Suppose that r i=1 tr(Mi ) = 0, where tr(A) denotes the trace of the matrix A. Prove that r M is the n × n zero matrix. i i=1

2.2

47th Anual William Lowell Putnam Competition, 1986

1. Find, with explanation, the maximum value of f (x) = x3 ? 3x on the set of all real numbers x satisfying x4 + 36 ≤ 13x2 . 2. What is the units (i.e., rightmost) digit of 1020000 ? 10100 + 3

2 3. Evaluate ∞ n=0 Arccot(n + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval 0 < θ ≤ π/2 with cot θ = t.

4. A transversal of an n × n matrix A consists of n entries of A, no two in the same row or column. Let f (n) be the number of n × n matrices A satisfying the following two conditions: (a) Each entry αi,j of A is in the set {?1, 0, 1}.

(b) The sum of the n entries of a transversal is the same for all transversals of A. An example of such a matrix A is ?1 0 ?1 ? ? A = ? 0 1 0 ?. 0 1 0

? ?

2.2. 47T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1986 Determine with proof a formula for f (n) of the form

n n f (n) = a1 bn 1 + a 2 b2 + a 3 b3 + a 4 ,

53

where the ai ’s and bi ’s are rational numbers. 5. Suppose f1 (x), f2 (x), . . . , fn (x) are functions of n real variables x = (x1 , . . . , xn ) with continuous second-order partial derivatives everywhere on Rn . Suppose further that there are constants cij such that ?fj ?fi ? = cij ?xj ?xi for all i and j , 1 ≤ i ≤ n, 1 ≤ j ≤ n. Prove that there is a function g (x) on Rn such that fi + ?g/?xi is linear for all i, 1 ≤ i ≤ n. (A linear function is one of the form a0 + a1 x1 + a2 x2 + · · · + an xn .) 6. Let a1 , a2 , . . . , an be real numbers, and let b1 , b2 , . . . , bn be distinct positive integers. Suppose that there is a polynomial f (x) satisfying the identity

n

(1 ? x) f (x) = 1 +

n

a i xb i .

i=1

Find a simple expression (not involving any sums) for f (1) in terms of b1 , b2 , . . . , bn and n (but independent of a1 , a2 , . . . , an ). 7. Inscribe a rectangle of base b and height h in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut o? by one base of the rectangle (with that side as the base of the triangle). For what value of h do the rectangle and triangle have the same area? 8. Prove that there are only a ?nite number of possibilities for the ordered triple T = (x ? y, y ? z, z ? x), where x, y, z are complex numbers satisfying the simultaneous equations x(x ? 1) + 2yz = y (y ? 1) + 2zx + z (z ? 1) + 2xy, and list all such triples T . 9. Let Γ consist of all polynomials in x with integer coe?cienst. For f and g in Γ and m a positive integer, let f ≡ g (mod m) mean that every coe?cient of f ? g is an integral multiple of m. Let n and p be positive integers with p prime. Given that f, g, h, r and s are in Γ with rf + sg ≡ 1 (mod p) and f g ≡ h (mod p), prove that there exist F and G in Γ with F ≡ f (mod p), G ≡ g (mod p), and F G ≡ h (mod pn ).

54

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

√ 10. For a positive real number r , let G(r ) be the minimum value of |r ? m2 + 2n2 | for all integers m and n. Prove or disprove the assertion that limr→∞ G(r ) exists and equals 0. 11. Let f (x, y, z ) = x2 + y 2 + z 2 + xyz . Let p(x, y, z ), q (x, y, z ), r (x, y, z ) be polynomials with real coe?cients satisfying f (p(x, y, z ), q (x, y, z ), r (x, y, z )) = f (x, y, z ). Prove or disprove the assertion that the sequence p, q, r consists of some permutation of ±x, ±y, ±z , where the number of minus signs is 0 or 2. 12. Suppose A, B, C, D are n × n matrices with entries in a ?eld F , satisfying the conditions that AB T andCD T are symmetric and AD T ? BC T = I . Here I is the n × n identity matrix, and if M is an n × n matrix, M T is its transpose. Prove that AT D + C T B = I .

2.3

48th Anual William Lowell Putnam Competition, 1987

x , + y2

1. Curves A, B, C and D are de?ned in the plane as follows: A = B = C = D = Prove that A ∩ B = C ∩ D . 2. The sequence of digits 123456789101112131415161718192021 . . . is obtained by writing the positive integers in order. If the 10n -th digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, de?ne f (n) to be m. For example, f (2) = 2 because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, f (1987). (x, y ) : x2 ? y 2 = (x, y ) : 2xy + x2

y =3 , x2 + y 2

(x, y ) : x3 ? 3xy 2 + 3y = 1 ,

(x, y ) : 3x2 y ? 3x ? y 3 = 0 .

2.3. 48T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1987 3. For all real x, the real-valued function y = f (x) satis?es y ? 2y + y = 2ex . (a) If f (x) > 0 for all real x, must f (x) > 0 for all real x? Explain. (b) If f (x) > 0 for all real x, must f (x) > 0 for all real x? Explain.

55

4. Let P be a polynomial, with real coe?cients, in three variables and F be a function of two variables such that P (ux, uy, uz ) = u2 F (y ? x, z ? x) for all real x, y, z, u, and such that P (1, 0, 0) = 4, P (0, 1, 0) = 5, and P (0, 0, 1) = 6. Also let A, B, C be complex numbers with P (A, B, C ) = 0 and |B ? A| = 10. Find |C ? A|. 5. Let G(x, y ) = ?y x , ,0 . x2 + 4y 2 x2 + 4y 2

Prove or disprove that there is a vector-valued function F (x, y, z ) = (M (x, y, z ), N (x, y, z ), P (x, y, z )) with the following properties: (i) M, N, P have continuous partial derivatives for all (x, y, z ) = (0, 0, 0); (ii) Curl F = 0 for all (x, y, z ) = (0, 0, 0); (iii) F (x, y, 0) = G(x, y ). 6. For each positive integer n, let a(n) be the number of zeroes in the base 3 representation of n. For which positive real numbers x does the series xa(n) 3 n=1 n converge? 7. Evaluate

4 2 ∞

ln(9 ? x) dx ln(9 ? x) + ln(x + 3)

.

56

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION 8. Let r, s and t be integers with 0 ≤ r , 0 ≤ s and r + s ≤ t. Prove that

s 0 t r

+

s 1 t r +1

+···+

s s t r +s

=

t+1 (t + 1 ? s)

t?s r

9. Let F be a ?eld in which 1 + 1 = 0. Show that the set of solutions to the equation x2 + y 2 = 1 with x and y in F is given by (x, y ) = (1, 0) and (x, y ) = r 2 ? 1 2r , r2 + 1 r2 + 1

where r runs through the elements of F such that r 2 = ?1. 10. Let (x1 , y1 ) = (0.8, 0.6) and let xn+1 = xn cos yn ? yn sin yn and yn+1 = xn sin yn + yn cos yn for n = 1, 2, 3, . . .. For each of limn→∞ xn and limn→∞ yn , prove that the limit exists and ?nd it or prove that the limit does not exist. 11. Let On be the n-dimensional vector (0, 0, · · · , 0). Let M be a 2n × n matrix of complex numbers such that whenever (z1 , z2 , . . . , z2n )M = On , with complex zi , not all zero, then at least one of the zi is not real. Prove that for arbitrary real numbers r1 , r2 , . . . , r2n , there are complex numbers w1 , w2 , . . . , wn such that w1 ? ? . ?? ? ? ? ? ? re ?M ? . . ? ?? = ? wn

? ? ?? ?

(Note: if C is a matrix of complex numbers, re(C ) is the matrix whose entries are the real parts of the entries of C .) 12. Let F be the ?eld of p2 elements, where p is an odd prime. Suppose S is a set of (p2 ? 1)/2 distinct nonzero elements of F with the property that for each a = 0 in F , exactly one of a and ?a is in S . Let N be the number of elements in the intersection S ∩ {2a : a ∈ S }. Prove that N is even.

r1 ? . . . ? ?. rn

?

2.4

49th Anual William Lowell Putnam Competition, 1988

1. Let R be the region consisting of the points (x, y ) of the cartesian plane satisfying both |x| ? |y | ≤ 1 and |y | ≤ 1. Sketch the region R and ?nd its area.

2.4. 49T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1988

57

2. A not uncommon calculus mistake is to believe that the product rule for derivatives 2 says that (f g ) = f g . If f (x) = ex , determine, with proof, whether there exists an open interval (a, b) and a nonzero function g de?ned on (a, b) such that this wrong product rule is true for x in (a, b). 3. Determine, with proof, the set of real numbers x for which

∞ n=1

1 1 csc ? 1 n n

x

converges. 4. (a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? (b) What if three is replaced by nine ? 5. Prove that there exists a unique function f from the set R+ of positive real numbers to R+ such that f (f (x)) = 6x ? f (x) and f (x) > 0 for all x > 0. 6. If a linear transformation A on an n-dimensional vector space has n + 1 eigenvectors such that any n of them are linearly independent, does it follow that A is a scalar multiple of the identity? Prove your answer. 7. A composite (positive integer) is a product ab with a and b not necessarily distinct integers in {2, 3, 4, . . .}. Show that every composite is expressible as xy + xz + yz + 1, with x, y, z positive integers. 8. Prove or disprove: If x and y are real numbers with y ≥ 0 and y (y + 1) ≤ (x + 1)2 , then y (y ? 1) ≤ x2 . 9. For every n in the set N = {1, 2, . . .} of positive integers, let rn be the minimum value √ of |c ? d 3| for all nonnegative integers c and d with c + d = n. Find, with proof, the smallest positive real number g with rn ≤ g for all n ∈ N. 10. Prove that if ∞ n=1 an is a convergent series of positive real numbers, then so is ∞ n/(n+1) ( a ) . n=1 n

58

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

11. For positive integers n, let Mn be the 2n + 1 by 2n + 1 skew-symmetric matrix for which each entry in the ?rst n subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of Mn . (According to one de?nition, the rank of a matrix is the largest k such that there is a k × k submatrix with nonzero determinant.) One may note that 0 ?1 1 ? M1 = ? 1 0 ?1 ? ? ?1 1 0 M2 =

? ? ? ? ? ? ? ? ? ? ?

0 ?1 ?1 1 1 ? 1 0 ?1 ?1 1 ? ? 1 1 0 ?1 ?1 ? ? ?1 1 1 0 ?1 ? ? ?1 ?1 1 1 0

12. Prove that there exist an in?nite number of ordered pairs (a, b) of integers such that for every positive integer t, the number at + b is a triangular number if and only if t is a triangular number. (The triangular numbers are the tn = n(n + 1)/2 with n in {0, 1, 2, . . .})

2.5

50th Anual William Lowell Putnam Competition, 1989

1. How many primes among the positive integers, written as usual in base 10, are alternating 1’s and 0’s, beginning and ending with 1? 2. Evaluate

a 0 0 b

emax{b

2 x2 ,a2 y 2 }

dy dx where a and b are positive.

3. Prove that if 11z 10 + 10iz 9 + 10iz ? 11 = 0, then |z | = 1. (Here z is a complex number and i2 = ?1.) 4. If α is an irrational number, 0 < α < 1, is there a ?nite game with an honest coin such that the probability of one player winning the game is α? (An honest coin is one . A game is for which the probability of heads and the probability of tails are both 1 2 ?nite if with probability 1 it must end in a ?nite number of moves.)

2.5. 50T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1989

59

5. Let m be a positive integer and let G be a regular (2m + 1)-gon inscribed in the unit circle. Show that there is a positive constant A, independent of m, with the following property. For any points p inside G there are two distinct vertices v1 and v2 of G such that 1 A | |p ? v1 | ? |p ? v2 | | < ? 3. m m Here |s ? t| denotes the distance between the points s and t. 6. Let α = 1 + a1 x + a2 x2 + · · · be a formal power series with coe?cients in the ?eld of two elements. Let

? ? ? ? ? ? ? ? ?

1

an =

if every block of zeros in the binary expansion of n has an even number of zeros in the block otherwise.

0

(For example, a36 = 1 because 36 = 1001002 and a20 = 0 because 20 = 101002 .) Prove that α3 + xα + 1 = 0. 7. A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, ?nd the probability that the point √ hit a b+c , is nearer to the center than to any edge. Express your answer in the form d where a, b, c, d are integers. 8. Let S be a non-empty set with an associative operation that is left and right cancellative (xy = xz implies y = z , and yx = zx implies y = z ).ssume that for every a in S the set {an : n = 1, 2, 3, . . .} is inite. Must S be a group? 9. Let f be a function on [0, ∞), di?erentiable and satisfying f (x) = ?3f (x) + 6f (2x) for x > 0. Assume that |f (x)| ≤ e? x for x ≥ 0 (so that f (x) tends rapidly to 0 as x increases). For n a non-negative integer, de?ne ?n =

∞ 0 √

xn f (x) dx

(sometimes called the nth moment of f ). a) Express ?n in terms of ?0 . b) Prove that the sequence {?n 3 } always converges, and that the limit is 0 only n! if ?0 = 0.

n

60

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. Can a countably in?nite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is ?nite? 11. Label the vertices of a trapezoid T (quadrilateral with two parallel sides) inscribed in the unit circle as A, B, C, D so that AB is parallel to CD and A, B, C, D are in counterclockwise order. Let s1 , s2 , and d denote the lengths of the line segments AB, CD , and OE , where E is the point of intersection of the diagonals of T , and O s2 is the center of the circle. Determine the least upper bound of s1 ? over all such T d for which d = 0, and describe allases, if any, in which it is attained. 12. Let (x1 , x2 , . . . xn ) be a point chosen at random from the n-dimensional region de?ned by 0 < x1 < x2 < · · · < xn < 1. Let f be a continuous function on [0, 1] with f (1) = 0. Set x0 = 0 and xn+1 = 1. Show that the expected value of the Riemann sum

n i=0

(xi+1 ? xi )f (xi+1 )

is 01 f (t)P (t) dt, where P is a polynomial of degree n, independent of f , with 0 ≤ P (t) ≤ 1 for 0 ≤ t ≤ 1.

2.6

51th Anual William Lowell Putnam Competition, 1990

T0 = 2, T1 = 3, T2 = 6, and for n ≥ 3, Tn = (n + 4)Tn?1 ? 4nTn?2 + (4n ? 8)Tn?3 . 2, 3, 6, 14, 40, 152, 784, 5168, 40576.

1. Let

The ?rst few terms are

Find, with proof, a formula for Tn of the form Tn = An + Bn , where {An } and {Bn } are well-known sequences. √ √ √ 2. Is 2 the limit of a sequence of numbers of the form 3 n ? 3 m (n, m = 0, 1, 2, . . .)? 3. Prove that any convex pentagon whose vertices (no three of which are collinear) have 5 integer coordinates must have area greater than or equal to 2 .

2.6. 51T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1990

61

4. Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? 5. If A and B are square matrices of the same size such that ABAB = 0, does it follow that BABA = 0? 6. If X is a ?nite set, let X denote the number of elements in X . Call an ordered pair (S, T ) of subsets of {1, 2, . . . , n} admissible if s > |T | for each s ∈ S , and t > |S | for each t ∈ T . How many admissible ordered pairs of subsets of {1, 2, . . . , 10} are there? Prove your answer. 7. Find all real-valued continuously di?erentiable functions f on the real line such that for all x, x (f (x))2 = [(f (t))2 + (f (t))2 ] dt + 1990.

0

8. Prove that for |x| < 1, |z | > 1, 1+ where Pj is

∞

(1 + xj )Pj = 0,

j =1

(1 ? z )(1 ? zx)(1 ? zx2 ) · · · (1 ? zxj ?1 ) . (z ? x)(z ? x2 )(z ? x3 ) · · · (z ? xj )

9. Let S be a set of 2 × 2 integer matrices whose entries aij (1) are all squares of integers and, (2) satisfy aij ≤ 200. Show that if S has more than 50387 (= 154 ? 152 ? 15 + 2) elements, then it has two elements that commute. 10. Let G be a ?nite group of order n generated by a and b. Prove or disprove: there is a sequence g1 , g2 , g3 , . . . , g2n such that (1) every element of G occurs exactly twice, and (2) gi+1 equals gi a or gi b for i = 1, 2, . . . , 2n. (Interpret g2n+1 as g1 .) 11. Is there an in?nite sequence a0 , a1 , a2 , . . . of nonzero real numbers such that for n = 1, 2, 3, . . . the polynomial pn (x) = a0 + a1 x + a2 x2 + · · · + an xn has exactly n distinct real roots?

62

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

12. Let S be a nonempty closed bounded convex set in the plane. Let K be a line and t a positive number. Let L1 and L2 be support lines for S parallel to K1 , and let L be the line parallel to K and midway between L1 and L2 . Let BS (K, t) be the band of points whose distance from L is at most (t/2)w , where w is the distance between L1 and L2 . What is the smallest t such that S∩

K

BS (K, t) = ?

for all S ? (K runs over all lines in the plane.)

2.7

52th Anual William Lowell Putnam Competition, 1991

1. A 2 × 3 rectangle has vertices as (0, 0), (2, 0), (0, 3), and (2, 3). It rotates 90 ? clockwise about the point (2, 0). It then otates 90? clockwise about the point (5, 0), then 90? clockwise about the point (7, 0), and ?nally, 90? clockwise about the point (10, 0). (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is (1,1). 2. Let A and B be di?erent n × n matrices with real entries. If A3 = B 3 and A2 B = B 2 A, can A2 + B 2 be nvertible? 3. Find all real polynomials p(x) of degree n ≥ 2 for which there exist real numbers r1 < r2 < · · · < rn such that (a) p(ri ) = 0, (b) p

ri +ri+1 2

i = 1, 2, . . . , n, and =0 i = 1, 2, . . . , n ? 1,

where p (x) denotes the derivative of p(x). 4. Does there exist an in?nite sequence of closed discs D1 , D2 , D3 , . . . in the plane, with centers c1 , c2 , c3 , . . ., respectively, such that (a) the ci have no limit point in the ?nite plane, (b) the sum of the areas of the Di is ?nite, and (c) every line in the plane intersects at least one of the Di ?

2.7. 52T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1991 5. Find the maximum value of

y 0

63

x4 + (y ? y 2 )2 dx

for 0 ≤ y ≤ 1. 6. Let A(n) denote the number of sums of positive integers a1 + a 2 + · · · + a r which add up to n with a1 > a 2 + a 3 , a2 > a 3 + a 4 , . . . ar?2 > ar?1 + ar , ar?1 > ar Let B (n) denote the number of b1 + b2 + · · · + bs which add up to n, with (b) each bi is in the sequence 1, 2, 4, . . . , gj , . . . de?ned by g1 = 1, g2 = 2, and gj = gj ?1 + gj ?2 + 1, and (c) if b1 = gk then every element in {1, 2, 4, . . . , gk } appears at least once as a bi . Prove that A(n) = B (n) for each n ≥ 1. (For example, A(7) = 5 because the relevant sums are 7, 6 + 1, 5 + 2, 4 + 3, 4 + 2 + 1, and B (7) = 5 because the relevant sums are 4 + 2 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.) (a) b1 ≥ b2 ≥ . . . ≥ bs ,

7. For each integer n ≥ 0, let S (n) = n ? m2 , where m is the greatest integer with m2 ≤ n. De?ne a sequence (ak )∞ k =0 by a0 = A and ak +1 = ak + S (ak ) for k ≥ 0. For what positive integers A is this sequence eventually constant? 8. Suppose f and g are non-constant, di?erentiable, real-valued functions de?ned on (?∞, ∞). Furthermore, suppose that for each pair of real numbers x and y , f (x + y ) = f (x)f (y ) ? g (x)g (y ), g (x + y ) = f (x)g (y ) + g (x)f (y ). If f (0) = 0, prove that (f (x))2 + (g (x))2 = 1 for all x. 9. Does there exist a real number L such that, if m and n are integers greater than L, then an m × n rectangle may be expressed as a nion of 4 × 6 and 5 × 7 rectangles, any two of which intersect at most along their boundaries?

64

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. Suppose p is an odd prime. Prove that

p j =0

p j

p+j ≡ 2p + 1 (mod p2 ). j

11. Let p be an odd prime and let Zp denote (the ?eld of) integers modulo p. How many elements are in the set {x2 : x ∈ Zp } ∩ {y 2 + 1 : y ∈ Zp }? 12. Let a and b be positive numbers. Find the largest number c, in terms of a and b, such that sinh u(1 ? x) sinh ux +b ax b1?x ≤ a sinh u sinh u for all u with 0 < |u| ≤ c and for all x, 0 < x < 1. (Note: sinh u = (eu ? e?u )/2.)

2.8

53th Anual William Lowell Putnam Competition, 1992

1. Prove that f (n) = 1 ? n is the only integer-valued function de?ned on the integers that satis?es the following conditions. (i) f (f (n)) = n, for all integers n; (ii) f (f (n + 2) + 2) = n for all integers n; (iii) f (0) = 1. 2. De?ne C (α) to be the coe?cient of x1992 in the power series about x = 0 of (1 + x)α . Evaluate 1992 1 1 C (?y ? 1) dy. 0 k =1 y + k 3. For a given positive integer m, ?nd all triples (n, x, y ) of positive integers, with n relatively prime to m, which satisfy (x2 + y 2 )m = (xy )n .

2.8. 53T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1992

65

4. Let f be an in?nitely di?erentiable real-valued function de?ned on the real numbers. If 1 n2 , n = 1, 2, 3, . . . , f = 2 n n +1 compute the values of the derivatives f (k) (0), k = 1, 2, 3, . . .. 5. For each positive integer n, let an = 0 (or 1) if the number of 1’s in the binary representation of n is even (or odd), respectively. Show that there do not exist positive integers k and m such that ak+j = ak+m+j = ak+m+2j , for 0 ≤ j ≤ m ? 1. 6. Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) 7. Let S be a set of n distinct real numbers. Let AS be the set of numbers that occur as averages of two distinct elements of S . For a given n ≥ 2, what is the smallest possible number of elements in AS ? 8. For nonnegative integers n and k , de?ne Q(n, k ) to be the coe?cient of xk in the expansion of (1 + x + x2 + x3 )n . Prove that

k

Q(n, k ) =

j =0

n j

n , k ? 2j

where a ≥ 0,

a b a b

is the standard binomial coe?cient. (Reminder: For integers a and b with =

a! b!(a?b)!

for 0 ≤ b ≤ a, with

a b

= 0 otherwise.)

9. For any pair (x, y ) of real numbers, a sequence (an (x, y ))n≥0 is de?ned as follows: a0 (x, y ) = x, (an (x, y ))2 + y 2 , an+1 (x, y ) = 2 Find the area of the region {(x, y )|(an (x, y ))n≥0 converges }.

for n ≥ 0.

66

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. Let p(x) be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with x3 ? x. Let d1992 dx1992 p(x) x3 ? x = f (x) g (x)

for polynomials f (x) and g (x). Find the smallest possible degree of f (x). 11. Let Dn denote the value of the (n ? 1) × (n ? 1) determinant

? ? ? ? ? ? ? ? ? ? ?

n≥2

3 1 1 1 . . .

1 4 1 1 . . .

1 1 5 1 . . .

1 1 1 6 . . .

··· ··· ··· ··· .. .

1 1 1 1 . . .

?

1 1 1 1 ··· n+1

Is the set

Dn n!

? ? ? ? ? ?. ? ? ? ?

bounded?

12. Let M be a set of real n × n matrices such that (ii) if A ∈ M and B ∈ M, then either AB ∈ M or ?AB ∈ M, but not both; (i) I ∈ M, where I is the n × n identity matrix;

(iii) if A ∈ M and B ∈ M, then either AB = BA or AB = ?BA; Prove that M contains at most n2 matrices.

(iv) if A ∈ M and A = I , there is at least one B ∈ M such that AB = ?BA.

2.9

54th Anual William Lowell Putnam Competition, 1993

1. The horizontal line y = c intersects the curve y = 2x ? 3x3 in the ?rst quadrant as in the ?gure. Find c so that the areas of the two shaded regions are equal. [Figure not included. The ?rst region is bounded by the y -axis, the line y = c and the curve; the other lies under the curve and above the line y = c between their two points of intersection.] 2. Let (xn )n≥0 be a sequence of nonzero real numbers such that x2 n ? xn?1 xn+1 = 1 for n = 1, 2, 3, . . .. Prove there exists a real number a such that xn+1 = axn ? xn?1 for all n ≥ 1.

2.9. 54T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1993

67

3. Let Pn be the set of subsets of {1, 2, . . . , n}. Let c(n, m) be the number of functions f : Pn → {1, 2, . . . , m} such that f (A ∩ B ) = min{f (A), f (B )}. Prove that

m

c(n, m) =

j =1

j n.

4. Let x1 , x2 , . . . , x19 be positive integers each of which is less than or equal to 93. Let y1 , y2 , . . . , y93 be positive integers each of which is less than or equal to 19. Prove that there exists a (nonempty) sum of some xi ’s equal to a sum of some yj ’s. 5. Show that

?10 ?100

x2 ? x x3 ? 3x + 1

2

dx +

1 11 1 101

x2 ? x x3 ? 3x + 1

2

dx +

11 10 101 100

x2 ? x x3 ? 3x + 1

2

dx

is a rational number. 6. The in?nite sequence of 2’s and 3’s 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, . . . as the property that, if one forms a second sequence that records the number of 3’s between successive 2’s, the result is identical to the iven sequence. Show that there exists a real number r such that, for ny n, the nth term of the sequence is 2 if and only if n = 1 + rm for some nonnegative integer m. (Note: xrf loor denotes the largest integer less than or equal to x.) 7. Find the smallest positive integer n such that for every integer m with 0 < m < 1993, there exists an integer k for which m k m+1 < < . 1993 n 1994 8. Consider the following game played with a deck of 2n cards numbered from 1 to 2n. The deck is randomly shu?ed and n cards are dealt to each of two players. Beginning with A, the players take turns discarding one of their remaining cards and announcing its number. The game ends as soon as the sum of the numbers on the discarded cards is divisible by 2n + 1. The last person to discard wins the game. Assuming optimal strategy by both A and B , what is the probability that A wins? 9. Two real numbers x and y are chosen at random in the interval (0,1) with respect to the uniform distribution. What is the probability that he closest integer to x/y is even? Express the answer in the form r + sπ , where r and s are rational numbers.

68

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. The function K (x, y ) is positive and continuous for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and the functions f (x) and g (x) are positive and continuous for 0 ≤ x ≤ 1. Suppose that for all x, 0 ≤ x ≤ 1,

1 0

f (y )K (x, y ) dy = g (x)

and

0

1

g (y )K (x, y ) dy = f (x).

Show that f (x) = g (x) for 9 ≤ x ≤ 1. 11. Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers. 12. Let S be a set of three, not necessarily distinct, positive integers. Show that one can transform S into a set containing 0 by a ?nite number of applications of the following rule: Select two of the three integers, say x and y , where x < y and replace them with 2x and y ? x.

2.10

55th Anual William Lowell Putnam Competition, 1994

1. Let (an ) be a sequence of positive reals such that, for all n, an ≤ a2n + a2n+1 . Prove that ∞ n=1 an diverges. 2. Find the positive value of m such that the area in the ?rst quadrant enclosed by the 2 + y 2 = 1, the x-axis, and the line y = 2x/3 is equal to the area in the ?rst ellipse x 9 2 + y 2 = 1, the y -axis, and the line y = mx. quadrant enclosed by the ellipse x 9 3. Prove that the points of an isosceles triangle of side length √ 1 annot be colored in four colors such that no two points at distance at least 2 ? 2 from each other receive the same color. 4. Let A and B be 2 × 2 matrices with integer entries such that each of A, A + B, A + 2B, A + 3B, A + 4B has an inverse with integer entries. Prove that the same must be true of A + 5B . 5. Let (rn ) be a sequence of positive reals with limit 0. Let S be the set of all numbers expressible in the form ri1 + . . . + ri1994 for positive integers i1 < i2 < . . . < i1994 . Prove that every interval (a, b) contains a subinterval (c, d) whose intersection with S is empty.

2.11. 56T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1995

69

6. Let f1 , . . . , f10 be bijections of the integers such that for every integer n, there exists a sequence i1 , . . . , ik for some k such that fi1 ? . . . ? fik (0) = n. Prove that if A is any nonempty ?nite set, there exist at most 512 sequences (e1 , . . . , e10 ) of zeroes and e1 e10 ones such that f1 ? . . . ? f10 maps A to A. (Here f 1 = f and f 0 means the identity function.) 7. Find all positive integers n such that |n ? m2 | ≤ 250 for exactly 15 nonnegative integers m. 8. Find all c such that the graph of the function x4 + 9x3 + cx2 + ax + b meets some line in four distinct points. 9. Let f (x) be a positive-valued function over the reals such that f (x) > f (x) for all x. For what k must there exist N such that f (x) > ekx for x > N ? 10. Let A be the matrix 3 2 and for positive integers n, de?ne dn as the greatest 4 2 common divisor of the entries of An ? I , where I = ((10)(01)). Prove that dn → ∞ as n → ∞.

11. Fix n a positive integer. For α real, de?ne fα (i) as the greatest integer less than or equal to αi, and write f k for the k -th iterate of f (i.e. f 1 = f and f k+1 = f ? f k ). k (n2 ) = n2 ? k for k = 1, . . . , n. Prove there exists α such that fαk (n2 ) = fα 12. Suppose a, b, c, d are integers with 0 ≤ a ≤ bleq 99, 0 ≤ c ≤ d ≤ 99. For any integer i, let ni = 101i + 1002i . Show that if na + nb is congruent to nc + nd mod 10100, then a = c and b = d.

2.11

56th Anual William Lowell Putnam Competition, 1995

1. Let S be a set of real numbers which is closed under multiplication (that is, if a and b are in S , then so is ab). Let T and U be disjoint subsets of S whose union is S . Given that the product of any three (not necessarily distinct) elements of T is in T and that the product of any three elements of U is in U , show that at least one of the two subsets T, U is closed under multiplication. 2. For what pairs (a, b) of positive real numbers does the improper integral

∞ b

√ √ x+a? x?

√ √ x ? x ? b dx

70 converge?

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

3. The number d1 d2 . . . d9 has nine (not necessarily distinct) decimal digits. The number e1 e2 . . . e9 is such that each of the nine 9-digit numbers formed by replacing just one of the digits di is d1 d2 . . . d9 by the corresponding digit ei (1 ≤ i ≤ 9) is divisible by 7. The number f1 f2 . . . f9 is related to e1 e2 . . . e9 is the same way: that is, each of the nine numbers formed by replacing one of the ei by the corresponding fi is divisible by 7. Show that, for each i, di ? fi is divisible by 7. [For example, if d1 d2 . . . d9 = 199501996, then e6 may be 2 or 9, since 199502996 and 199509996 are multiples of 7.] 4. Suppose we have a necklace of n beads. Each bead is labeled with an integer and the sum of all these labels is n ? 1. Prove that we can cut the necklace to form a string whose consecutive labels x1 , x2 , . . . , xn satisfy

k i=1

xi ≤ k ? 1

for k = 1, 2, . . . , n.

5. Let x1 , x2 , . . . , xn be di?erentiable (real-valued) functions of a single variable f which satisfy dx1 = a11 x1 + a12 x2 + · · · + a1n xn dt dx2 = a21 x1 + a22 x2 + · · · + a2n xn dt . . . . . . dxn = an1 x1 + an2 x2 + · · · + ann xn dt for some constants aij > 0. Suppose that for all i, xi (t) → 0 as t → ∞. Are the functions x1 , x2 , . . . , xn necessarily linearly dependent? 6. Suppose that each of n people writes down the numbers 1,2,3 in random order in one column of a 3 × n matrix, with all orders equally likely and with the orders for di?erent columns independent of each other. Let the row sums a, b, c of the resulting matrix be rearranged (if necessary) so that a ≤ b ≤ c. Show that for some n ≥ 1995, it is at least four times as likely that both b = a + 1 and c = a + 2 as that a = b = c. 7. For a partition π of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let π (x) be the number of elements in the part containing x. Prove that for any two partitions π and π , there are two distinct numbers x and y in {1, 2, 3, 4, 5, 6, 7, 8, 9} such that π (x) = π (y ) and π (x) = π (y ). [A partition of a set S is a collection of disjoint subsets (parts) whose union is S .]

2.12. 57T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1996

71

8. An ellipse, whose semi-axes have lengths a and b, rolls without slipping on the curve . How are a, b, c related, given that the ellipse completes one revolution y = c sin x a when it traverses one period of the curve? 9. To each positive integer with n2 decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for 8 6 n = 2, to the integer 8617 we associate det = 50. Find, as a function of n, 1 7 the sum of all the determinants associated with n2 -digit integers. (Leading digits are assumed to be nonzero; for example, for n = 2, there are 9000 determinants.) 10. Evaluate

8

2207 ?

√ a+b c , d

1 . 1 2207 ? 2207 ?... where a, b, c, d are integers.

Express your answer in the form

11. A game starts with four heaps of beans, containing 3,4,5 and 6 beans. The two players move alternately. A move consists of taking either a) one bean from a heap, provided at least two beans are left behind in that heap, or b) a complete heap of two or three beans. The player who takes the last heap wins. To win the game, do you want to move ?rst or second? Give a winning strategy. 12. For a positive real number α, de?ne S (α) = { nα : n = 1, 2, 3, . . .}. Prove that {1, 2, 3, . . .} cannot be expressed as the disjoint union of three sets S (α), S (β ) and S (γ ). [As usual, x is the greatest integer ≤ x.]

2.12

57th Anual William Lowell Putnam Competition, 1996

1. Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle.

72

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION 2. Let C1 and C2 be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find, with proof, the locus of all points M for which there exists points X on C1 and Y on C2 such that M is the midpoint of the line segment XY . 3. Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses o?ered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course. 4. Let S be the set of ordered triples (a, b, c) of distinct elements of a ?nite set A. Suppose that (a) (a, b, c) ∈ S if and only if (b, c, a) ∈ S ;

(b) (a, b, c) ∈ S if and only if (c, b, a) ∈ / S;

(c) (a, b, c) and (c, d, a) are both in S if and only if (b, c, d) and (d, a, b) are both in S.

Prove that there exists a one-to-one function g from A to R such that g (a) < g (b) < g (c) implies (a, b, c) ∈ S . Note: R is the set of real numbers. 5. If p is a prime number greater than 3 and k = 2p/3 , prove that the sum p p p + +···+ 1 2 k of binomial coe?cients is divisible by p2 . 6. Let c > 0 be a constant. Give a complete description, with proof, of the set of all continuous functions f : R → R such that f (x) = f (x2 + c) for all x ∈ R. Note that R denotes the set of real numbers. 7. De?ne a sel?sh set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of {1, 2, . . . , n} which are minimal sel?sh sets, that is, sel?sh sets none of whose proper subsets is sel?sh. 8. Show that for every positive integer n, 2n ? 1 e

2n?1 2

2n + 1 < 1 · 3 · 5 · · · (2n ? 1) < e

2n+1 2

.

9. Given that {x1 , x2 , . . . , xn } = {1, 2, . . . , n}, ?nd, with proof, the largest possible value, as a function of n (with n ≥ 2), of x1 x2 + x2 x3 + · · · + xn?1 xn + xn x1 .

2.13. 58T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1997 10. For any square matrix A, we can de?ne sin A by the usual power series: (?1)n sin A = A2n+1 . n=0 (2n + 1)! Prove or disprove: there exists a 2 × 2 matrix A with real entries such that sin A = 1 1996 0 1 .

∞

73

11. Given a ?nite string S of symbols X and O , we write ?(S ) for the number of X ’s in S minus the number of O ’s. For example, ?(XOOXOOX ) = ?1. We call a string S balanced if every substring T of (consecutive symbols of) S has ?2 ≤ ?(T ) ≤ 2. Thus, XOOXOOX is not balanced, since it contains the substring OOXOO . Find, with proof, the number of balanced strings of length n. 12. Let (a1 , b1 ), (a2 , b2 ), . . . , (an , bn ) be the vertices of a convex polygon which contains the origin in its interior. Prove that there exist positive real numbers x and y such that (a1 , b1 )xa1 y b1 + (a2 , b2 )xa2 y b2 + · · · + (an , bn )xan y bn = (0, 0)

2.13

58th Anual William Lowell Putnam Competition, 1997

1. A rectangle, HOM F , has sides HO = 11 and OM = 5. A triangle ABC has H as the intersection of the altitudes, O the center of the circumscribed circle, M the midpoint of BC , and F the foot of the altitude from A. What is the length of BC ? 2. Players 1, 2, 3, . . . , n are seated around a table, and each has a single penny. Player 1 passes a penny to player 2, who then passes two pennies to player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an in?nite set of numbers n for which some player ends up with all n pennies. 3. Evaluate

∞ 0

x?

x5 x7 x3 + ? +··· 2 2·4 2·4·6

1+

x2 x4 x6 + + + · · · dx. 22 22 · 42 22 · 42 · 62

74

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION 4. Let G be a group with identity e and φ : G → G a function such that φ(g1 )φ(g2 )φ(g3 ) = φ(h1 )φ(h2 )φ(h3 ) whenever g1 g2 g3 = e = h1 h2 h3 . Prove that there exists an element a ∈ G such that ψ (x) = aφ(x) is a homomorphism (i.e. ψ (xy ) = ψ (x)ψ (y ) for all x, y ∈ G). 5. Let Nn denote the number of ordered n-tuples of positive integers (a1 , a2 , . . . , an ) such that 1/a1 + 1/a2 + . . . + 1/an = 1. Determine whether N10 is even or odd. 6. For a positive integer n and any real number c, de?ne xk recursively by x0 = 0, x1 = 1, and for k ≥ 0, cxk+1 ? (n ? k )xk . xk+2 = k+1 Fix n and then take c to be the largest value for which xn+1 = 0. Find xk in terms of n and k , 1 ≤ k ≤ n. 7. Let {x} denote the distance between the real number x and the nearest integer. For each positive integer n, evaluate Fn =

6n?1 m=1

min({

m m }, { }). 6n 3n

(Here min(a, b) denotes the minimum of a and b.) 8. Let f be a twice-di?erentiable real-valued function satisfying f (x) + f (x) = ?xg (x)f (x), where g (x) ≥ 0 for all real x. Prove that |f (x)| is bounded. 9. For each positive integer n, write the sum n m=1 1/m in the form pn /qn , where pn and qn are relatively prime positive integers. Determine all n such that 5 does not divide qn . 10. Let am,n denote the coe?cient of xn in the expansion of (1 + x + x2 )m . Prove that for all [integers] k ≥ 0,

2k 3

0≤ 11. Prove that for n ≥ 2, n terms 22

···2

i=0

(?1)i ak?i,i ≤ 1.

n ? 1 terms ≡ 22

···2

(mod n).

2.14. 59T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1998

75

12. The dissection of the 3–4–5 triangle shown below (into four congruent right triangles similar to the original) has diameter 5/2. Find the least diameter of a dissection of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)

2.14

59th Anual William Lowell Putnam Competition, 1998

1. A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? 2. Let s be any arc of the unit circle lying entirely in the ?rst quadrant. Let A be the area of the region lying below s and above the x-axis and let B be the area of the region lying to the right of the y -axis and to the left of s. Prove that A + B depends only on the arc length, and not on the position, of s. 3. Let f be a real function on the real line with continuous third derivative. Prove that there exists a point a such that f (a) · f (a) · f (a) · f (a) ≥ 0. 4. Let A1 = 0 and A2 = 1. For n > 2, the number An is de?ned by concatenating the decimal expansions of An?1 and An?2 from left to right. For example A3 = A2 A1 = 10, A4 = A3 A2 = 101, A5 = A4 A3 = 10110, and so forth. Determine all n such that 11 divides An . 5. Let F be a ?nite collection of open discs in R2 whose union contains a set E ? R2 . Show that there is a pairwise disjoint subcollection D1 , . . . , Dn in F such that E ? ∪n j =1 3Dj . Here, if D is the disc of radius r and center P , then 3D is the disc of radius 3r and center P . 6. Let A, B, C denote distinct points with integer coordinates in R2 . Prove that if (|AB | + |BC |)2 < 8 · [ABC ] + 1 then A, B, C are three vertices of a square. Here |XY | is the length of segment XY and [ABC ] is the area of triangle ABC .

76

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION 7. Find the minimum value of (x + 1/x)6 ? (x6 + 1/x6 ) ? 2 (x + 1/x)3 + (x3 + 1/x3 ) for x > 0. 8. Given a point (a, b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a, b), one on the x-axis, and one on the line y = x. You may assume that a triangle of minimum perimeter exists. 9. let H be the unit hemisphere {(x, y, z ) : x2 + y 2 + z 2 = 1, z ≥ 0}, C the unit circle {(x, y, 0) : x2 + y 2 = 1}, and P the regular pentagon inscribed in C . Determine the surface area of that portion of H lying over the planar region inside P , and write your answer in the form A sin α + B cos β , where A, B, α, β are real numbers.

10. Find necessary and su?cient conditions on positive integers m and n so that

mn?1 i=0

(?1)

i/m + i/n

= 0.

11. Let N be the positive integer with 1998 decimal digits, all of them 1; that is, N = 1111 · · · 11. Find the thousandth digit after the decimal point of

√

N. √ n3 + an2 + bn + c

12. Prove that, for any integers a, b, c, there exists a positive integer n such that is not an integer.

2.15

60th Anual William Lowell Putnam Competition, 1999

1. Find polynomials f (x),g (x), and h(x), if they exist, such that for all x, ?1 if x < ?1 3x + 2 if ?1 ≤ x ≤ 0 |f (x)| ? |g (x)| + h(x) = ? ? ?2x + 2 if x > 0.

? ? ?

2.15. 60T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1999

77

2. Let p(x) be a polynomial that is nonnegative for all real x. Prove that for some k , there are polynomials f1 (x), . . . , fk (x) such that

k

p(x) =

j =1

(fj (x))2 .

3. Consider the power series expansion

∞ 1 = a n xn . 1 ? 2x ? x2 n=0

Prove that, for each integer n ≥ 0, there is an integer m such that

2 a2 n + an+1 = am .

4. Sum the series

m2 n . m m n m=1 n=1 3 (n3 + m3 )

∞

∞

5. Prove that there is a constant C such that, if p(x) is a polynomial of degree 1999, then 1 |p(0)| ≤ C |p(x)| dx.

?1

6. The sequence (an )n≥1 is de?ned by a1 = 1, a2 = 2, a3 = 24, and, for n ≥ 4, an =

2 6a2 n?1 an?3 ? 8an?1 an?2 . an?2 an?3

Show that, for all n, an is an integer multiple of n. 7. Right triangle ABC has right angle at C and ∠BAC = θ ; the point D is chosen on AB so that |AC | = |AD | = 1; the point E is chosen on BC so that ∠CDE = θ . The perpendicular to BC at E meets AB at F . Evaluate limθ→0 |EF |. 8. Let P (x) be a polynomial of degree n such that P (x) = Q(x)P (x), where Q(x) is a quadratic polynomial and P (x) is the second derivative of P (x). Show that if P (x) has at least two distinct roots then it must have n distinct roots. 9. Let A = {(x, y ) : 0 ≤ x, y < 1}. For (x, y ) ∈ A, let S (x, y ) =

1 ≤m ≤2 2 n

xm y n ,

78

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION where the sum ranges over all pairs (m, n) of positive integers satisfying the indicated inequalities. Evaluate

(x,y )→(1,1),(x,y )∈A

lim

(1 ? xy 2 )(1 ? x2 y )S (x, y ).

10. Let f be a real function with a continuous third derivative such that f (x), f (x), f (x), f (x) are positive for all x. Suppose that f (x) ≤ f (x) for all x. Show that f (x) < 2f (x) for all x. 11. For an integer n ≥ 3, let θ = 2π/n. Evaluate the determinant of the n × n matrix I + A, where I is the n × n identity matrix and A = (ajk ) has entries ajk = cos(jθ + kθ ) for all j, k . 12. Let S be a ?nite set of integers, each greater than 1. Suppose that for each integer n there is some s ∈ S such that gcd(s, n) = 1 or gcd(s, n) = s. Show that there exist s, t ∈ S such that gcd(s, t) is prime.

2.16. 61ST ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2000

79

2.16

61st Anual William Lowell Putnam Competition, 2000

∞ j =0

1. Let A be a positive real number. What are the possible values of x0 , x1 , . . . are positive numbers for which ∞ j =0 xj = A?

x2 j , given that

2. Prove that there exist in?nitely many integers n such that n, n + 1, n + 2 are each the sum of the squares of two integers. [Example: 0 = 02 + 02 , 1 = 02 + 12 , 2 = 12 + 12 .] 3. The octagon P1 P2 P3 P4 P5 P6 P7 P8 is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon P1 P3 P5 P7 is a square of area 5, and the polygon P2 P4 P6 P8 is aectangle of area 4, ?nd the maximum possible area of the octagon. 4. Show that the improper integral

B →∞ 0

lim

B

sin(x) sin(x2 ) dx

converges. 5. Three distinct points with integer coordinates lie in the plane on a circle of radius r > 0. Show that two of these points are separated by a distance of at least r 1/3 . 6. Let f (x) be a polynomial with integer coe?cients. De?ne a sequence a0 , a1 , . . . of integers such that a0 = 0 and an+1 = f (an ) for all n ≥ 0. Prove that if there exists a positive integer m for which am = 0 then either a1 = 0 or a2 = 0. 7. Let aj , bj , cj be integers for 1 ≤ j ≤ N . Assume for each j , at least one of aj , bj , cj is odd. Show that there exist integers r , s, t such that raj + sbj + tcj is odd for at least 4N/7 values of j , 1 ≤ j ≤ N . 8. Prove that the expression gcd(m, n) n n m is an integer for all pairs of integers n ≥ m ≥ 1. 9. Let f (t) = N j =1 aj sin(2πjt), where each aj is real and aN is not equal to 0. Let Nk kf denote the number of zeroes (including multiplicities) of d . Prove that dtk N0 ≤ N1 ≤ N2 ≤ · · · and lim Nk = 2N.

k →∞

[Editorial clari?cation: only zeroes in [0, 1) should be counted.]

80

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION

10. Let f (x) be a continuous function such that f (2x2 ? 1) = 2xf (x) for all x. Show that f (x) = 0 for ?1 ≤ x ≤ 1. 11. Let S0 be a ?nite set of positive integers. We de?ne ?nite sets S1 , S2 , . . . of positive integers as follows: the integer a is in Sn+1 if and only if exactly one of a ? 1 or a is in Sn . Show that there exist in?nitely many integers N for which SN = S0 ∪ {N + a : a ∈ S0 }. 12. Let B be a set of more than 2 n distinct points with coordinates of the form (±1, ±1, . . . , ±1) in n-dimensional space with n ≥ 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle.

n+1

2.17

62nd Anual William Lowell Putnam Competition, 2001

1. Consider a set S and a binary operation ?, i.e., for each a, b ∈ S , a ? b ∈ S . Assume (a ? b) ? a = b for all a, b ∈ S . Prove that a ? (b ? a) = b for all a, b ∈ S . 2. You have coins C1 , C2 , . . . , Cn . For each k , Ck is biased so that, when tossed, it has probability 1/(2k +1) of falling heads. If the n coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function of n. 3. For each integer m, consider the polynomial Pm (x) = x4 ? (2m + 4)x2 + (m ? 2)2 . For what values of m is Pm (x) the product of two non-constant polynomials with integer coe?cients? 4. Triangle ABC has an area 1. Points E, F, G lie, respectively, on sides BC , CA, AB such that AE bisects BF at point R, BF bisects CG at point S , and CG bisects AE at point T . Find the area of the triangle RST . 5. Prove that there are unique positive integers a, n such that an+1 ? (a + 1)n = 2001. 6. Can an arc of a parabola inside a circle of radius 1 have a length greater than 4? 7. Let n be an even positive integer. Write the numbers 1, 2, . . . , n2 in the squares of an n × n grid so that the k -th row, from left to right, is (k ? 1)n + 1, (k ? 1)n + 2, . . . , (k ? 1)n + n.

2.18. 63RD ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2002

81

Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares. 8. Find all pairs of real numbers (x, y ) satisfying the system of equations

1 x 1 x

+ ?

1 2y 1 2y

= (x2 + 3y 2 )(3x2 + y 2 ) = 2(y 4 ? x4 ). √ n. Evaluate

9. For any positive integer n, let n denote the closest integer to 2 n + 2? n . 2n n=1

∞

10. Let S denote the set of rational numbers di?erent from {?1, 0, 1}. De?ne f : S → S by f (x) = x ? 1/x. Prove or disprove that

∞ n=1

f (n) (S ) = ?,

where f (n) denotes f composed with itself n times. 11. Let a and b be real numbers in the interval (0, 1/2), and let g be a continuous realvalued function such that g (g (x)) = ag (x) + bx for all real x. Prove that g (x) = cx for some constant c. 12. Assume that (an )n≥1 is an increasing sequence of positive real numbers such that lim an /n = 0. Must there exist in?nitely many positive integers n such that an?i + an+i < 2an for i = 1, 2, . . . , n ? 1?

2.18

63rd Anual William Lowell Putnam Competition, 2002

1 xk ?1

1. Let k be a ?xed positive integer. The n-th derivative of where Pn (x) is a polynomial. Find Pn (1).

has the form

Pn (x) (xk ?1)n+1

2. Given any ?ve points on a sphere, show that some four of them must lie on a closed hemisphere.

82

CHAPTER 2. WILLIAM LOWELL PUTNAM COMPETITION 3. Let n ≥ 2 be an integer and Tn be the number of non-empty subsets S of {1, 2, 3, . . . , n} with the property that the average of the elements of S is an integer. Prove that Tn ? n is always even. 4. In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3 × 3 matrix. Player 0 counters with a 0 in a vacant position, and play continues in turn until the 3 × 3 matrix is completed with ?ve 1’s and four 0’s. Player 0 wins if the determinant is 0 and player 1 wins otherwise. Assuming both players pursue optimal strategies, who will win and how? 5. De?ne a sequence by a0 = 1, together with the rules a2n+1 = an and a2n+2 = an + an+1 for each integer n ≥ 0. Prove that every positive rational number appears in the set an?1 :n≥1 = an 1 1 2 1 3 , , , , ,... . 1 2 1 3 2

6. Fix an integer b ≥ 2. Let f (1) = 1, f (2) = 2, and for each n ≥ 3, de?ne f (n) = nf (d), where d is the number of base-b digits of n. For which values of b does 1 n=1 f (n) converge? 7. Shanille O’Keal shoots free throws on a basketball court. She hits the ?rst and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her ?rst 100 shots? 8. Consider a polyhedron with at least ?ve faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each player, in turn, signs his or her name on a previously unsigned face. The winner is the player who ?rst succeeds in signing three faces that share a common vertex. Show that the player who signs ?rst will always win by playing as well as possible. 9. Show that, for all integers n > 1, 1 1 1 < ? 1? 2ne e n

n ∞

<

1 . ne

2.18. 63RD ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2002

83

10. An integer n, unknown to you, has been randomly chosen in the interval [1, 2002] with uniform probability. Your objective is to select n in an odd number of guesses. After each incorrect guess, you are informed whether n is higher or lower, and you must guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than 2/3. 11. A palindrome in base b is a positive integer whose base-b digits read the same backwards and forwards; for example, 2002 is a 4-digit palindrome in base 10. Note that 200 is not a palindrome in base 10, but it is the 3-digit palindrome 242 in base 9, and 404 in base 7. Prove that there is an integer which is a 3-digit palindrome in base b for at least 2002 di?erent values of b. 12. Let p be a prime number. Prove that the determinant of the matrix

?

x y z ? p p y zp ? ? x ? 2 2 p p p2 x y z is congruent modulo p to a product of polynomials of the form ax + by + cz , where a, b, c are integers. (We say two integer polynomials are congruent modulo p if corresponding coe?cients are congruent modulo p.)

?

Chapter 3 Asiatic Paci?c Mathematical Olympiads

3.1 1st Asiatic Paci?c Mathematical Olympiad, 1989

S = x1 + x2 + · · · + x n . (1 + x1 )(1 + x2 ) · · · (1 + xn ) ≤ 1 + S + 2. Prove that the equation 6(6a2 + 3b2 + c2 ) = 5n2 has no solutions in integers except a = b = c = n = 0. 3. Let A1 , A2 , A3 be three points in the plane, and for convenience, let A4 = A1 , A5 = A2 . For n = 1, 2, and 3, suppose that Bn is the midpoint of An An+1 , and suppose that Cn is the midpoint of An Bn . Suppose that An Cn+1 and Bn An+2 meet at Dn , and that An Bn+1 and Cn An+2 meet at En . Calculate the ratio of the area of triangle D1 D2 D3 to the area of triangle E1 E2 E3 . 4. Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least (m ? n ) 4 4m · 3n 84

2

1. Let x1 , x2 , . . . , xn be positive real numbers, and let Prove that

Sn S2 S3 + +···+ . 2! 3! n!

3.2. 2N D ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1990 triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S .

85

5. Determine all functions f from the reals to the reals for which (1) f (x) is strictly increasing, (2) f (x) + g (x) = 2x for all real x, where g (x) is the composition inverse function to f (x). (Note: f and g are said to be composition inverses if f (g (x)) = x and g (f (x)) = x for all real x.)

3.2

2nd Asiatic Paci?c Mathematical Olympiad, 1990

1. Given triagnle ABC , let D , E , F be the midpoints of BC , AC , AB respectively and let G be the centroid of the triangle. For each value of ∠BAC , how many non-similar triangles are there in which AEGF is a cyclic quadrilateral? 2. Let a1 , a2 , . . . , an be positive real numbers, and let Sk be the sum of the products of a1 , a2 , . . . , an taken k at a time. Show that Sk Sn?k for k = 1, 2, . . . , n ? 1. 3. Consider all the triangles ABC which have a ?xed base AB and whose altitude from C is a constant h. For which of these triangles is the product of its altitudes a maximum? 4. A set of 1990 persons is divided into non-intersecting subsets in such a way that: (a) No one in a subset knows all the others in the subset, (b) Among any three persons in a subset, there are always at least two who do not know each other, and (c) For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them. a Prove that within each subset, every person has the same number of acquaintances. b Determine the maximum possible number of subsets. n ≥ k

2

a1 a2 · · · an

86

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS Note: It is understood that if a person A knows person B , then person B will know person A; an acquaintance is someone who is known. Every person is assumed to know one’s self. 5. Show that for every integer n ≥ 6, there exists a convex hexagon which can be dissected into exactly n congruent triangles.

3.3

3rd Asiatic Paci?c Mathematical Olympiad, 1991

1. Let G be the centroid of triangle ABC and M be the midpoint of BC . Let X be on AB and Y on AC such that the points X , Y , and G are collinear and XY and BC are parallel. Suppose that XC and GB intersect at Q and Y B and GC intersect at P . Show that triangle M P Q is similar to triangle ABC . 2. Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane. Can you ?nd a special case with exactly 1991 red points? 3. Let a1 , a2 , . . . , an , b1 , b2 , . . . , bn be positive real numbers such that a1 + a2 + · · · + an = b1 + b2 + · · · + bn . Show that a2 a2 a2 a1 + a 2 + · · · + a n 1 2 n + +···+ ≥ a1 + b 1 a2 + b 2 an + b n 2

4. During a break, n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of n for which eventually, perhaps after many rounds, all children will have at least one candy each. 5. Given are two tangent circles and a point P on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point P .

3.4

4th Asiatic Paci?c Mathematical Olympiad, 1992

1. A triangle with sides a, b, and c is given. Denote by s the semiperimeter, that is

3.5. 5T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1993

87

b+c s = a+2 . Construct a triangle with sides s ? a, s ? b, and s ? c. This process is repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated inde?nitely?

2. In a circle C with centre O and radius r , let C1 , C2 be two circles with centres O1 , O2 and radii r1 , r2 respectively, so that each circle Ci is internally tangent to C at Ai and so that C1 , C2 are externally tangent to each other at A.Prove that the three lines OA, O1 A2 , and O2 A1 are concurrent. 3. Let n be an integer such that n > 3. Suppose that we choose three numbers from the set {1, 2, . . . , n}. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than n/2, then thealues of these combinations are all distinct. √ (b) Let p be a prime number such that p ≤ n. Show that the number of ways of choosing three numbers so that the smallest one is p and the values of the combinations are not all distinct is precisely the number of positive divisors of p ? 1. 4. Determine all pairs (h, s) of positive integers with the following property: If one draws h horizontal lines and another s lines which satisfy: i they are not horizontal, ii no two of them are parallel, iii no three of the h + s lines are concurrent, then the number of regions formed by these h + s lines is 1992. 5. Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

3.5

5th Asiatic Paci?c Mathematical Olympiad, 1993

1. Let ABCD be a quadrilateral such that all sides have equal length and angle ABC is 60 deg. Let l be a line passing through D and not intersecting the quadrilateral (except at D ). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF . Prove that CA2 = CM · CE .

88

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS 2. Find the total number of di?erent integer values the function f (x) = [x] + [2x] + [ takes for real numbers x with 0 ≤ x ≤ 100. 3. Let f (x) = an xn + an?1 xn?1 + · · · + a0 and g (x) = cn+1 xn+1 + cn xn + · · · + c0 5x ] + [3x] + [4x] 3

be non-zero polynomials with real coe?cients such that g (x) = (x + r )f (x) for some real number r . If a = max(|an |, . . . , |a0 |) and c = max(|cn+1 |, . . . , |c0 |), prove that a ≤ n + 1. c 4. Determine all positive integers n for which the equation xn + (2 + x)n + (2 ? x)n = 0 has an integer as a solution. 5. Let P1 , P2 , . . . , P1993 = P0 be distinct points in the xy -plane with the following properties: i both coordinates of Pi are integers, for i = 1, 2, . . . , 1993; ii there is no point other than Pi and Pi+1 on the line segment joining Pi with Pi+1 whose coordinates are both integers, for i = 0, 1, . . . , 1992. Prove that for some i, 0 ≤ i ≤ 1992, there exists a point Q with coordinates (qx , qy ) on the line segment joining Pi with Pi+1 such that both 2qx and 2qy are odd integers.

3.6

6th Asiatic Paci?c Mathematical Olympiad, 1994

i For all x, y ∈ R, f (x) + f (y ) + 1 ≥ f (x + y ) ≥ f (x) + f (y ) iii ?f (?1) = f (1) = 1. ii For all x ∈ [0, 1), f (0) ≥ f (x),

1. Let f : R → R be a function such that:

3.7. 7T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1995 Find all such functions f .

89

2. Given a nondegenerate triangle ABC , with circumcentre O , orthocentre H , and circumradius R, prove that |OH | < 3R. 3. Let n be an integer of the form a2 + b2 , where a and b are relatively prime integers √ and such that if p is a prime, p ≤ n, then p divides ab. Determine all such n. 4. Is there an in?nite set of points in the plane such that no three points are collinear, and the distance between any two points is rational? 5. You are given three lists A, B , and C . List A contains the numbers of the form 10k in base 10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: A 10 100 1000 . . . B 1010 1100100 1111101000 . . . C 20 400 13000 . . .

Prove that for every integer n > 1, there is exactly one number in exactly one of the lists B or C that has exactly n digits.

3.7

7th Asiatic Paci?c Mathematical Olympiad, 1995

1. Determine all sequences of real numbers a1 , a2 , . . . , a1995 which satisfy: 2 an ? (n ? 1) ≥ an+1 ? (n ? 1), for n = 1, 2, . . . 1994, and √ 2 a1995 ? 1994 ≥ a1 + 1.

2. Let a1 , a2 , . . . , an be a sequence of integers with values between 2 and 1995 such that: i Any two of the ai ’s are realtively prime, ii Each ai is either a prime or a product of primes. Determine the smallest possible values of n to make sure that the sequence will contain a prime number.

90

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS 3. Let P QRS be a cyclic quadrilateral such that the segments P Q and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S . Determine the set A of points of tangency of circles in these two sets. 4. Let C be a circle with radius R and centre O , and S a ?xed point in the interior of C . Let AA and BB be perpendicular chords through S . Consider the rectangles SAM B , SBN A , SA M B , and SB N A. Find the set of all points M , N , M , and N when A moves around the whole circle. 5. Find the minimum positive integer k such that there exists a function f from the set Z of all integers to {1, 2, . . . k } with the property that f (x) = f (y ) whenever |x ? y | ∈ {5, 7, 12}.

3.8

8th Asiatic Paci?c Mathematical Olympiad, 1996

1. Let ABCD be a quadrilateral AB = BC = CD = DA. Let M N and P Q be two segments perpendicular to the diagonal BD and such that the distance between them is d > BD/2, with M ∈ AD , N ∈ DC , P ∈ AB , and Q ∈ BC . Show that the perimeter of hexagon AM N CQP does not depend on the position of M N and P Q so long as the distance between them remains constant. 2. Let m and n be positive integers such that n ≤ m. Prove that 2n n! ≤ (m + n)! ≤ (m2 + m)n (m ? n)!

3. Let P1 , P2 , P3 , P4 be four points on a circle, and let I1 be the incentre of the triangle P2 P3 P4 ; I2 be the incentre of the triangle P1 P3 P4 ; I3 be the incentre of the triangle P1 P2 P4 ; I4 be the incentre of the triangle P1 P2 P3 . Prove that I1 , I2 , I3 , I4 are the vertices of a rectangle. 4. The National Marriage Council wishes to invite n couples to form 17 discussion groups under the following conditions: (a) All members of a group must be of the same sex; i.e. they are either all male or all female. (b) The di?erence in the size of any two groups is 0 or 1. (c) All groups have at least 1 member. (d) Each person must belong to one and only one group.

3.9. 9T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1997 Find all values of n, n ≤ 1996, for which this is possible. Justify your answer. 5. Let a, b, c be the lengths of the sides of a triangle. Prove that √ √ √ √ √ √ a+b?c+ b+c?a+ c+a?b≤ a+ b+ c , and determine when equality occurs.

91

3.9

9th Asiatic Paci?c Mathematical Olympiad, 1997

S =1+ 1 1+

1 3

1. Given

+

1 1+

1 3

+

1 6

+···+

1+

1 3

where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. k = n(n + 1)/2 for n = 1, 2, . . . , 1996). Prove that S > 1001. 2. Find an integer n, where 100 ≤ n ≤ 1997, such that 2n + 2 n is also an integer. 3. Let ABC be a triangle inscribed in a circle and let la = mb mc ma , lb = , lc = , Ma Mb Mc

1 + +···+

1 6

1 1993006

where ma , mb , mc are the lengths of the angle bisectors (internal to the triangle) and Ma , Mb , Mc are the lengths of the angle bisectors extended until they meet the circle. Prove that la lb lc + + ≥ 3, 2 2 sin A sin B sin2 C and that equality holds i? ABC is an equilateral triangle. 4. Triangle A1 A2 A3 has a right angle at A3 . A sequence of points is now de?ned by the following iterative process, where n is a positive integer. From An (n ≥ 3), a perpendicular line is drawn to meet An?2 An?1 at An+1 . (a) Prove that if this process is continued inde?nitely, then one and only one point P is interior to every triangle An?2 An?1 An , n ≥ 3.

92

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS (b) Let A1 and A3 be ?xed points. By considering all possible locations of A2 on the plane, ?nd the locus of P . 5. Suppose that n people A1 , A2 , . . ., An , (n ≥ 3) are seated in a circle and that Ai has ai objects such that a1 + a2 + · · · + an = nN, where N is a positive integer. In order that each person has the same number of objects, each person Ai is to give or to receive a certain number of objects to or from its two neighbours Ai?1 and Ai+1 . (Here An+1 means A1 and An means A0 .) How should this redistribution be performed so that the total number of objects transferred is minimum?

3.10

10th Asiatic Paci?c Mathematical Olympiad, 1998

1. Let F be the set of all n?tuples (A1 , . . . , An ) such that each Ai is a subset of {1, 2, . . . , 1998}. Let |A| denote the number of elements of the set A. Find:

(A1 ,...,An )∈F

|A1 ∪ · · · ∪ An |

2. Show that for any positive integers a and b, (36a + b)(a + 36b) can not be a power of 2. 3. Let a, b, c be positive real numbers. Pruve that: 1+ a b 1+ b c 1+ c a+b+c ≥2 1+ √ 3 a abc

4. Let ABC be a triangle and D the foot of the altitude from A. Let E and F lie on a line through D such that AE is perpendicular to BC , AF is perpendicular to CF , and E and F are di?erent from D . Let M and N be the midpoint of the segments BC and EF , respectively. Prove that AN is perpendicular to N M . 5. √ Find the largest integer n such that n is divisible by all positive integers less than 3 n.

3.11. 11T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1999

93

3.11

11th Asiatic Paci?c Mathematical Olympiad, 1999

1. Find the smallest positive integer n with the following property: there does not exist an arithmetic progression of 1999 real numbers containing exactly n integers. 2. Let a1 , a2 , . . . be a sequence of real numbers satisfying ai+j ≤ ai + aj for all i, j = 1, 2, . . .. Prove that a2 a3 an a1 + + +···+ ≥ an 2 3 n for each positive integer n. 3. Let Γ1 and Γ2 be two circles intersecting at P and Q. The common tangent, closer to P , of Γ1 and Γ2 touches Γ1 at A and Γ2 at B . The tangent of Γ1 at P meets Γ2 at C , which is di?erent from P , and the extension of AP meets BC at R. Prove that the circumcircle of triangle P QR is tangent to BP and BR. 4. Determine all pairs (a, b) of integers with the property that the numbers a2 + 4b and b2 + 4a are both perfect squares. 5. Let S be a set of 2n + 1 points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of S on its circumference, n ? 1 points in its interior and n ? 1 points in its exterior. Prove that the number of good circles has the same parity as n.

3.12

12th Asiatic Paci?c Mathematical Olympiad, 2000

x3 i 2 1 ? 3 x i + 3xi i=0 xi = i 101

101

1. Compute the sum S= for

2. Given the following triangular arrangement of circles:

94

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS

Each of the numbers 1, . . . , 9 is to be written into one of these circles, so that each circle contain exactly one of these numbers and: i the sum of the four numbers on each side of the triangle are equal; ii the sum of the squares of the four numbers on each side of the triangle are equal. Find all ways in which this can be done. 3. Let ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectively, at A meet the side BC . Let Q and P be the point in which the perpendicular at N to N A meets M A and BA, respectively, and O the point in which the perpendicular at P to BA meets AN produced. Prove that QO is perpendicular to BC . 4. Let n, k be given positive integers with n > k . Prove that n! 1 nn nn < · < n + 1 k k (n ? k )n?k k ! (n ? k )! k k (n ? k )n?k 5. Given a permutation (a0 , a1 , . . . , an ) of the sequence 0, 1, . . . , n. A transposition of ai with aj is called legal if ai = 0 for i > 0, and ai?1 + 1 = aj . The permutation (a0 , . . . , an ) is called regular if after a number of transpositions it becomes (1, 2, . . . , n, 0). For which numbers n is the permutation (1, n, n ? 1, . . . , 3, 2, 0) regular?

3.13

13th Asiatic Paci?c Mathematical Olympiad, 2001

1. For a positive integer n let S (n) be the sum of digits in the decimal representation of n. Any positive integer obtained by removing several (at least one) digits from the

3.14. 14T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 2002

95

right-hand end of the decimal representation of n is called a stump of n. Let T (n) be the sum of all stumps of n. Prove that n = S (n) + 9T (n). 2. Find the largest positive integer N so that the number of integers in the set {1, 2, . . . , N } which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). 3. Let two equal regular n-gons S and T be located in the plane such that their intersection is a 2n-gon (n ≥ 3). The sides of the polygon S are coloured in red and the sides of T in blue. Prove that the sum of the lengths of the blue sides of the polygon S ∩ T is equal to the sum of the lengths of its red sides. 4. A point in the plane with a cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coe?cients such that their graphs do not contain any mixed point. 5. Find the greatest integer n, such that there are n +4 points A, B , C , D , X1 , . . . , Xn in the plane with AB = CD that satisfy the following condition: for each i = 1, 2, . . . , n triangles ABXi and CDXi are equal.

3.14

14th Asiatic Paci?c Mathematical Olympiad, 2002

1. Let a1 , a2 , a3 , . . . , an be a sequence of non-negative integers, where n is a positive integer. Let a1 + a 2 + · · · + a n An = . n Prove that a1 !a2 ! . . . an ! ≥ ( An !)n , where An is the greatest integer less than or equal to An , and a! = 1 × 2 × · · · × a for a ≥ 1 (and 0! = 1). When does equality hold? 2. Find all positive integers a and b such that a2 + b b2 ? a are both integers. 3. Let ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles ABP and ACQ are acute. Let R be and b2 + a a2 ? b

96

CHAPTER 3. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of CBP and BCQ such that triangle T RS is equilateral. 4. Let x, y, z be positive numbers such that 1 1 1 + + = 1. x y z Show that √ x + yz + √ √ √ √ √ √ y + zx + z + xy ≥ xyz + x + y + z.

5. Let R denote the set of all real numbers. Find all functions f from R to R satisfying: (i) there are only ?nitely many s in R such that f (s) = 0, and (ii) f (x4 + y ) = x3 f (x) + f (f (y )) for all x, y in R.

3.15

15th Asiatic Paci?c Mathematical Olympiad, 2003

p(x) = x8 ? 4x7 + 7x6 + ax5 + bx4 + cx3 + dx2 + ex + f

1. Let a, b, c, d, e, f be real numbers such that the polynomial

factorises into eight linear factors x ? xi , with xi > 0 for i = 1, 2, . . . , 8. Determine all possible values of f . 2. Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines 1 and 2 , which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect 1 at E and F respectively. Also, sides CB and CD intersect 2 at G and H respectively. Let the perimeters of AEF and CGH be m1 and m2 respectively. Prove that no matter how the square was placed, m1 + m2 remains constant. 3. Let k ≥ 14 be an integer, and let pk be the largest prime number which is strictly less than k . You may assume that pk ≥ 3k/4. Let n be a composite integer. Prove: (a) if n = 2pk , then n does not divide (n ? k )! ; (b) if n > 2pk , then n divides (n ? k )! 4. Let a, b, c be the sides of a triangle, with a + b + c = 1, and let n ≥ 2 be an integer. Show that √ n √ √ √ 2 n n n a n + b n + bn + c n + cn + a n < 1 + . 2

3.15. 15T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 2003

97

5. Given two positive integers m and n, ?nd the smallest positive integer k such that among any k people, either there are 2m of them who form m pairs of mutually acquainted people or there are 2n of them forming n pairs of mutually unacquainted people.

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