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香港培正中学第10届数学邀请赛决赛(中2组)


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

決賽(中二組) Final Event (Secondary 2) 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.

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

1.

小琪做家課時要計算一道形如 a ? b ? c 的算式(其中 a、b、c 是大於 1 的整?)。小琪 忘?要先乘除後加減,得出的結果比正確答案大? 2011。求 c。 In a homework problem Angel had to compute an expression of the form a ? b ? c where a, b, c are integers greater than 1. She forgot that multiplication and division should precede addition and subtraction and ended up with a result which is 2011 greater than the correct answer. Find c.

2.

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?

3.

某 n 位?的?字之和是 2011。求 n 的最小可能值。 The sum of digits of an n-digit number is 2011. Find the smallest possible value of n.

4.

在 ?ABC 中,D 是 A 到 BC 的垂足。? AB : AC ? 1: 2 , DB ? 3 且 DC ? 9 ,求 DA。 In ?ABC, D is the foot of the perpendicular from A to BC. If AB : AC ? 1: 2 , DB ? 3 and DC ? 9 , find DA. B

A

C D

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

5.

ABC 是個三角形,它的面積是 12。P 是三角形內的一點。已知 X、Y 和 Z 分別是 P 繞 A、B 和 C 點旋轉 180? 後的影像。求 ?XYZ 的面積。 ABC is a triangle of area 12. P is a point in the triangle. It is known that X, Y and Z are the images of P when rotated about A, B and C by 180? respectively. Find the area of ?XYZ.

2

6.

某班有 5 名學生,學號分別為 1 至 5。現每名學生可與其他同學成組,亦可自成一組, 但規定?名學號?續的學生?能同組。那麼共有多少種?同的分組方法? In a class there are 5 students, numbered 1 to 5. A student can now form groups with others or form a group by him/herself, subject to the restriction that two students whose class numbers are consecutive cannot be in the same group. How many different ways of grouping are there?

7.

已知 a、b、c 和 d 都是正整?,其中 a ? b ? c ? d 。? a ? b 、 a ? c 、 a ? d 、 b ? c 、 b ? d 和 c ? d 這?個?當中剛好有五個是質?,求 a 的最小可能值。 It is known that a, b, c and d are positive integers where a ? b ? c ? d . If exactly five of the six integers a ? b , a ? c , a ? d , b ? c , b ? d and c ? d are prime, find the smallest possible value of a.

8.

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 ?

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. 設 x1 、 x2 、 … 、 x100 為 介 乎 0.5 和 0.75 之 間 的 實 ? ( 包 括 0.5 和 0.75 ) 。 求 x1 (1 ? x2 ) ? x2 (1 ? x3 ) ? x3 (1 ? x4 ) ? ? ? x99 (1 ? x100 ) ? x100 (1 ? x1 ) 的最大可能值。 Let x1 , x2 , …, x100 be real numbers between 0.5 and 0.75 inclusive. Find the greatest possible value of x1 (1 ? x2 ) ? x2 (1 ? x3 ) ? x3 (1 ? x4 ) ? ? ? x99 (1 ? x100 ) ? x100 (1 ? x1 ) .

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11. ?把 564 ? 1 寫成二進制,其末尾有多少個?? How many ending zeros are there when 564 ? 1 is written in binary notation?

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. 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?

14. 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?)
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15. 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)。小權觀看? n 場比賽後,發現所有可能的分?都已經在計分牌上出現。求 n 的最 小可能值。(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). After watching n matches, Donald found that all possible scores have already appeared on the scoreboard. Find the smallest possible value of n. (Note: We regard a : b and b : a to be different scores if a ? b .)

16. 求滿足以下條件的五位正整?的?目: ? ? ? ? ? 每個?字皆?是 0。 該五位?可被 5 整除。 ?把最後一位?字刪去,所得的四位?可被 4 整除。 ?把最後?位?字刪去,所得的三位?可被 3 整除。 ?把最後三位?字刪去,所得的?位?可被 2 整除。

Find the number of 5-digit positive integers satisfying the following conditions: ? ? ? ? ? Every digit is non-zero. The 5-digit number is divisible by 5. If the last digit is removed, the resulting 4-digit number is divisible by 4. If the last two digits are removed, the resulting 3-digit number is divisible by 3. If the last three digits are removed, the resulting 2-digit number is divisible by 2.

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

17. 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?的最大可能值。 In the addition shown, each letter represents a different digit from 0 to 9. Find the greatest possible value of the five-digit number represented by PCIMC.
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T E N T H ? P C I M C 1 1 0 4 0 2

18. 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.)

19. 在一個重組?子遊戲中,?加者需要把 6 張分別寫上「?」、「學」、「很」、 「有」、「趣」和「味」的卡片重新排?。遊戲的計分方法如下:正確的排?是「?學 很有趣味」,在重新排?卡片後,我們把 6 張排?後的卡片分成最長的?續正確段(即 ?續地在正確排?中出現的卡片),每個由 k 張卡片組成的最長?續正確段可得 2 k 分。?如:如果排?是「味有趣?學很」,則最長的?續正確段分別是「味」、「有 趣」和「?學很」,因此得分是 21 ? 22 ? 23 ? 14 。?把卡片隨意排?,則最有可能得到 的分?是 S,且有 n 個?同的排?可以得到 S 分。求 n。 In a sentence reconstruction game, players had to rearrange 6 cards labelled ‘mathematics’, ‘is’, ‘a’, ‘very’, ‘interesting’ and ‘subject’. The score is computed as follows. With ‘mathematics is a very interesting subject’ being the correct order, the 6 rearranged cards are divided into maximal consecutive correct segments (i.e. consecutive cards which appear in the correct order). Each maximal consecutive correct segment consisting of k cards is worth 2 k points. For instance, for the rearrangement ‘subject very interesting mathematics is a’, the maximal consecutive correct segments are ‘subject’, ‘very interesting’ and ‘mathematics is a’, and hence the score is 21 ? 22 ? 23 ? 14 . By a random rearrangement of the cards, the most probable score is S with n different rearrangements leading to such score. Find n.

6

20. ?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

When each of the numbers

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

全卷完 END OF PAPER

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