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香港培正中学第10届数学邀请赛决赛(中3组)


第十屆培正?學邀請賽 10th Pui Ching Invitational Mathematics Competition

決賽(中三組) Final Event (Secondary 3) 時限:2 小時 Time allowed: 2 hours

?賽者須知: Instructions to Contestants: (a

) 本卷共設 20 題,總分為 100 分。 There are 20 questions in this paper and the total score is 100. (b) 除特別指明外,本卷內的所有?均為十進制。 Unless otherwise stated, all numbers in this paper are in decimal system. (c) 除特別指明外,所有答案須以?字的真確值表達,並化至最簡。?接受近似值。 Unless otherwise stated, all answers should be given in exact numerals in their simplest form. No approximation is accepted. (d) 把所有答案填在答題紙指定的空位上。毋須呈交計算步驟。 Put your answers on the space provided on the answer sheet. You are not required to hand in your steps of working. (e) ?得使用計算機。 The use of calculators is not allowed. (f) 本卷的附圖?一定依比?繪成。 The diagrams in this paper are not necessarily drawn to scale.

1

第 1 至第 4 題,每題 3 分。 Questions 1 to 4 each carries 3 marks.

1.

某?可寫成三個?續正整?之積,且可被 5 整除。該?的最小可能值是甚麼? A number can be written as the product of three consecutive positive integers and is divisible by 5. What is the smallest possible value of the number?

2.

在一個排球比賽中,開始時的計分牌顯示 0 : 0 ,每球勝出的一方可得 1 分,先得到 25 分的一方勝出(?設「刁時」,即如果打成 24 : 24 平手,再勝出一球的一方?會勝 出)。計分牌顯示的分?有多少個?同的可能?(註:? a ? b ,則 a : b 和 b : a 視為 ?同的分?。) In a volleyball match, the scoreboard initially shows 0 : 0 . Each time the winner scores 1 point, and whichever side gets 25 points first wins (no ‘deuce’ is played, i.e. when the score is 24 : 24 , the side getting the next point wins). How many different scores are possible on the scoreboard? (Note: We regard a : b and b : a to be different scores if a ? b .)

3.

現有?個容?分別為 3 公升和 5 公升的瓶子和一條水喉。開始時,?個瓶子都是空的。 在每個步驟中,我們可以選擇進?以下操作的其中一項: ? ? ? ?用水喉把一個空的瓶子或一個未盛滿水的瓶子注滿。 把一個瓶子中的水倒掉。 把一個瓶子中的水倒到另一個瓶子,直至所有水已倒進另一瓶子或另一瓶子已滿。

?要使得 5 公升的瓶子內盛有剛好 4 公升水,問最少要經過多少個步驟? There are two bottles with capacities 3 litres and 5 litres respectively, as well as a tap. In the beginning, both bottles are empty. In each step, we can perform one of the following operations: ? ? ? Fill an empty or a partially-filled bottle full of water using the water tap. Empty a bottle. Pour the water in one bottle into another bottle, until the other bottle is full or all water has been poured into the other bottle.

What is the minimum number of steps needed in order to have exactly 4 litres of water in the 5-litre bottle?

2

4.

顯恩把 20110402 的?字重新排?,得到一個?同的八位?,這個八位?跟原?相差 n。求 n 的最小可能值。 Joseph rearranged the digits of 20110402 to form a different eight-digit number which differs from the original number by n. Find the smallest possible value of n.

第 5 至第 8 題,每題 4 分。 Questions 5 to 8 each carries 4 marks.

5.

某扇形的半徑是 2,周界是 8。求它的面積。 A sector has radius 2 and perimeter 8. Find its area.

6.

如果一個 12 小時時鐘的鐘面上沒有?字,那麼當我們看著時鐘的鏡像時,會看出另一 個時間,?如:在 2 時看時鐘的鏡像會看出 10 時,?者相差? 4 小時(這?相差 4 小 時是指最接近的 10 時和 2 時,?如同一天的上午 10 時和下午 2 時?相差 4 小時,下 同)。在一天中,實際時間和鏡像時間相差剛好 2.5 小時的情況會出現多少次? If there are no numerical labels on a 12-hour clock, we will see another time by looking at the mirror image of the clock. For instance, at 2 o’clock the mirror image will look like 10 o’clock. The difference between the two times is 4 hours (referring to the interval between the closest 10 o’clock and 2 o’clock, for instance 10 am and 2 pm of the same day, same for below). In a day, how many instances are there such that the difference between the actual time and the time in the mirror image is exactly 2.5 hours?

7.

設 n 為 2011 位? 999…99。在 n 4 的?字中,有多少個是 9? Let n be the 2011-digit number 999…99. How many 9’s are there in the digits of n 4 ?

8.

設 A 是 18 位?、B 是 24 位?。?先寫下 A 再寫下 B,?可得到一個 42 位? C,當中有 k 個?同的?字。?先寫下 B 再寫下 A,所得的 42 位?亦是 C。求 k 所有可能值之和。 Let A be a 18-digit number and B be a 24-digit number. If one first writes A and then writes B, one gets a 42-digit number C, which consists of k different digits. It turns out that if one first writes B and then writes A, one also gets the 42-digit number C. Find the sum of all possible values of k.

3

第 9 至第 12 題,每題 5 分。 Questions 9 to 12 each carries 5 marks.

9.

?一個正整?可寫成 k 2 ? k (其中 k 是正整?)的形式,則我們稱它為「好?」。? 如:因為 62 ? 6 ? 42 ,故此 42 是「好?」。在首 2011 個正整?中,有多少個可寫成? 個「好?」之差? A positive integer is said to be ‘good’ if it can be expressed in the form k 2 ? k for some positive integer k. For instance, since 62 ? 6 ? 42 , we say that 42 is ‘good’. How many of the first 2011 positive integers can be expressed as the difference between two ‘good’ numbers?

10. 某次測驗有三名學生小文、小華和小強?加。測驗共設 n 題,分別佔 1、2、…、n 分, 每名考生在每題均取得 0 分或該題的全部分?。已知 ? ? ? ? ? ? 小文答對??續的 3 題,其他題目答錯; 小華答對??續的 4 題,其他題目答錯; 小強答對??續的 7 題,其他題目答錯; 小文、小華和小強的分?相同; 在其中一題,三人當中剛好有?人答對;且 在最後一題,三人當中剛好有一人答對。

求 n。 Three students, Raymond, Steven and Tony, took a test. In the test there were n questions, worth 1, 2, …, n marks respectively. In each question a candidate either got zero or all the marks allocated. Suppose that ? ? ? ? ? ? Raymond answered three consecutive questions correctly and the rest wrongly; Steven answered four consecutive questions correctly and the rest wrongly; Tony answered seven consecutive questions correctly and the rest wrongly; Raymond, Steven and Tony got the same marks; a question was correctly answered by exactly two of the three students; and the last question was correctly answered by exactly one of the three students.

Find n.

4

11. 2011 個?續正奇?之和是個?方?。? k 是當中最小的一個奇?,求 k 的最小可能值。 The sum of 2011 consecutive odd positive integers is a cubic number. If k is the smallest of these odd numbers, find the smallest possible value of k.

12. 在四邊形 ABCD 中, AB ? 4 、 CD ? 6 ,且 B 和 D 都 是直角。X 和 Y 分別是 AD 和 BC 上的點,使得 AX 和 CY 的長?均為整?。?四邊形 AXCY 的面積是 2011,則 AX 的長?有多少個?同的可能值? In quadrilateral ABCD, AB ? 4 , CD ? 6 and both B and D are right angles. X and Y are points on AD and BC respectively such that the lengths of AX and CY are both integers. If quadrilateral AXCY has area 2011, how many different possible values are there for the length of AX?

D X A

B

Y

C

第 13 至第 16 題,每題 6 分。 Questions 13 to 16 each carries 6 marks.

13. 在一個遊戲中,?加者需把 5 個球射進 5 條垂直的坑道。如果 5 個球最終成一直線,? 加者?勝出。(下圖顯示?四個?子,當中只有第一個和第四個勝出。)如果射?某? 目的球後已經肯定無法勝出,那麼遊戲?即結束,?加者?能再射餘下的球。?每個球 都必定會射進其中一條坑道,且射進各坑道的機會均等,求遊戲結束時射?的球的總? 期望值。 In a game, the player has to shoot 5 balls into one of the 5 vertical tracks, and wins if the 5 balls turn out to be collinear. (The figure below shows four examples, among which only the first and fourth outcomes are winning.) If the player is certain not to win after shooting a certain number of balls, the game ends immediately without shooting the remaining balls. If each ball must get into one of the tracks with equal probability, find the expected value of the number of balls shot when the game ends.

5

14. 某三角形的三條高分別長 2011、402 和 n,其中 n 是正整?。問 n 有多少個?同的可能 值? The three altitudes of a triangle have lengths 2011, 402 and n, where n is a positive integer. How many different possible values of n are there?

15. 在所示的算式中,每個字母代表一個由 0 至 9 的?同?字。求 PCIMC 所代表的五位?的最小可能值。 In the addition shown, each letter represents a different digit from 0 to 9. Find the smallest possible value of the five-digit number represented by PCIMC.

T E N T H ? P C I M C 1 1 0 4 0 2

16. 設 a1 ? 40 ,並對正整? n 定義
?an ? 1 n ? ? an ?1 ? ?an ? 1 n ? ? ?2011
n ??

? an 2 ? 2011 ? an 2 ? 2011 ? an 2 ? 2011

求 lim an 。(換?話?,當 n 很大時, an 會趨近甚麼??) Let a1 ? 40 . For positive integer n, we define
? an ? 1 n ? ? an ?1 ? ?an ? 1 n ? ? ?2011
n ??

if an 2 ? 2011 if an 2 ? 2011 if an 2 ? 2011

Find lim an . (In other words, to what number will an approach when n is large?)

6

第 17 至第 20 題,每題 7 分。 Questions 17 to 20 each carries 7 marks.

17. 在一個遊戲中,11 人圍圈而坐,另外有 11 張卡紙,其中 8 張是空白的,另外 3 張則分 別寫上「A」、「B」和「C」。開始時,每人隨機獲發一張卡紙。在每個回合中,每人 都會把手上的卡紙傳給右方的人,然後主持人會要求知道「A」、「B」和「C」三張卡 紙分別在誰人手上者舉手。結果在首三個回合中均沒有人舉手,而在第四回合中則有一 人舉手。?第五和第?回合分別有 x 人和 y 人舉手,求乘積 xy。(各人都?會看到別人 手上的卡紙,而且各人都是聰明的,即有足夠資?時?能作出推?。) In a game, 11 people sat in a circle. There were 11 cards, 8 of which were blank, and the numbers ‘A’, ‘B’ and ‘C’ were written on the other 3 respectively. One card was distributed to each person at random at the beginning. In each round, each person passed the card to the person on his right, and then the players would be asked to raise his hands if he could tell which people were holding the cards with ‘A’, ‘B’ and ‘C’ written respectively. It turned out that nobody raised hands during the first three rounds, while one person raised hands during the fourth round. During the fifth and sixth rounds, there were x and y people raising hands respectively. Find the product xy. (The people could not see the cards held by others. They are also intelligent, so that deductions can be made whenever sufficient information is available.)

18. ?把
和。

7293 ? 13 7293 ? 23 7293 ? 7283 、 、…、 分別寫成最簡分?,求所有分子之 7293 ? 7283 7293 ? 7273 7293 ? 13

7293 ? 13 7293 ? 23 7293 ? 7283 , , …, is written as a fraction When each of the numbers 7293 ? 7283 7293 ? 7273 7293 ? 13 in the lowest term, find the sum of all the numerators.

19. 圖中,ABC 是面積為 1 的三角形。D、E 是 AB 上的 點,F、G 則是 AC 上的點,使得 AD = DE = EB 和 AF = FG = GC。求 BF、BG、CD 和 CE 圍成的區域 的面積。 In the figure, ABC is a triangle with area 1. D, E are points on AB while F, G are points on AC such that AD = DE = EB and AF = FG = GC. Find the area of the region bounded by BF, BG, CD and CE.
E B

A D F G C

7

20. 陳先生、?先生、張先生和黃先生分別是教師、警察、醫生和?師,而他們的妻子則分 別是?師、教師、警察和醫生。這四對夫婦?加?一個象棋比賽,賽會以抽籤形式把八 人分成二人一組對賽,方法如下:把八個分別寫上八人的名字的球放進一個袋子,然後 逐一抽出,第一位被抽出的與第二位被抽出的對賽,第三位被抽出的與第四位被抽出的 對賽,如此?推。惟出現以下任何一種情況時,則最近一個抽出的球無效,須放回袋子 內重新抽出一個球: (1) 抽中?夫婦或?名從事同一職業的人對賽 (2) 在餘下的抽籤過程中,肯定會出現?夫婦或?名從事同一職業的人對賽(?如:抽 出第?個球後,袋子中只剩下代表陳先生和陳太太的球)
求抽籤過程中需要重新抽球的概?。

Mr Chan, Mr Lee, Mr Cheung and Mr Wong are a teacher, a policeman, a doctor and a lawyer respectively. Their wives are a lawyer, a teacher, a policeman and a doctor respectively. These four couples joined a chess competition. The organiser divided the eight participants into four groups of two to play against each other, as follows. Eight balls with the names of the eight contestants were put into a bag and then drawn one by one. The person first drawn would play against the one drawn second; the person drawn third would play against the one drawn fourth, and so on. However, if any of the following situations occurs, the latest ball drawn would be invalid and it would be put back into the bag and a ball had to be drawn again: (1) The draw requires a couple or two people with the same occupation to play against each other. (2) During the remaining draw, the situation where a couple or two people with the same occupation playing against each other would definitely occur (e.g. only the balls representing Mr Chan and Mrs Chan were left after the sixth ball was drawn). Find the probability that a ball has to be redrawn during the process of the draw.

全卷完 END OF PAPER

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