# 第54届国际数学奥林匹克(IMO2013)第2天试题(英文)

between respectively. diameter of be an acute triangle with orthocenter and . The points and , and let be a point on the side and , , is a

are the feet of the altitudes drawn from , and is a point such that , and are collinear.

is the circumcircle of triangle . Similarly,

is the circumcircle of triangle , and

is a point such that

is a diameter of

. Show that the points

5. Let

be the set of all rational numbers greater than zero. Let

be a function

satisfying the following conditions:

(i) (ii)

for all for all

, , such that .

(iii) There exists a rational number

Show that

for all

. equally spaced points marked on it. such that each label is used

6. Let

be an integer, and consider a circle with

Consider all labellings of these points with the numbers

exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels with , the chord joining the points labelled and . and does not

intersect the chord joining the points labelled

Let

be the number of beautiful labellings and let and

be the number of ordered pairs . Prove that

of

positive integers such that

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