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Thailand International Mathematical Olympiad 2015 泰國國際數學競賽 2015 Secondary 3 Past Paper Booklet 中學三年級 試題集 考生須知： Instructions to Contestants: 1.

本卷包括 試題 乙份，試題紙不可取走。 Each contestant should have ONE Question-Answer Book which CANNOT be taken away.

2.

本卷共 5 個範疇，每範疇有 5 題，共 25 題，每題 4 分，總分 100 分，答錯不扣分。 There are 5 exam areas and 5 questions in each exam area. There are a total of 25 questions in this Question-Answer Book. Each carries 4 marks. Total score is 100 marks. No points are deducted for incorrect answers.

3.

請將答案寫在 答題紙 上。 All answers should be written on ANSWER SHEET.

4.

比賽期間，不得使用計算工具。 NO calculators can be used during the contest.

5.

本卷中所有圖形不一定依比例繪成。 All figures in the paper are not necessarily drawn to scale.

6.

比賽完畢時，本試題會被收回。 This Question-Answer Book will be collected at the end of the contest. 本試題不可取走。 THIS Question-Answer Book CANNOT BE TAKEN AWAY. 未得監考官同意，切勿翻閱試題，否則參賽者將有可能被取消資格。 DO NOT turn over this Question-Answer Book without approval of the examiner. Otherwise, contestant may be DISQUALIFIED.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 填空題（第 1 至 25 題）（每題 4 分，答錯及空題不扣分） Open-Ended Questions (1st ~25th) (4 points for correct answer, no penalty point for wrong answer) Logical Thinking 邏輯思維 1.

A box contains 100 coloured balls: 30 blue, 30 red, 20 green and 20 yellow. Sherry takes some balls from the box without looking at the colours of the balls. What is the least number of balls that she must take so that she has 25 balls with same colour? 有一箱顏色球共有 100 個：30 個藍色、30 個紅色、20 個綠色及 20 個黃色。雪莉需要在不看球的顏 色的情況下從箱中取出顏色球，那麼她必須最少取出多少個球才能讓她取得 25 個相同顏色的球？

2.

According to the pattern shown below, what is the next number? 按以下規律，問下一個數是甚麼？ 2、9、28、65、126、217、344、…

3.

4.

1 2 2 2 2 . Find the integral part of     50 51 52 69 1 2 的整數部分。 求 2 2 2     50 51 52 69 There are six teams named A, B, C, D, E and F, participating a tournament. In 5 days, each team will play one game in each day. They play another team once in the tournament. So there are 3 tournaments every day. Given that: 1) Team A wins Team B on the first day. 2) Team C is defeated by Team D on the second day. 3) Team E wins Team A on the third day. 4) Team B wins Team C on the fourth day. Which team does Team B play with on the fifth day？ 有 A、B、C、D、E、F 六個籃球隊參加循環賽，賽程總共五天，每隊每天出賽一次，五天裡各與不同 的隊伍比賽一場，所以每天都有三場比賽，已知： 1) 第一天 A 隊贏了 B 隊 2) 第二天 C 隊輸給了 D 隊 3) 第三天 E 隊贏了 A 隊 4) 第四天 B 隊贏了 C 隊 請問，B 隊在第五天與哪一隊比賽？

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 5.

There are 7 people arguing which day of week is today Their statements are quoted as below: A: Yesterday was Wednesday. B: Tomorrow will not be Tuesday. C: Tomorrow will be Wednesday. D: The day after tomorrow will be Tuesday. E: Today is Tuesday. F: Today is neither Monday, Tuesday nor Sunday. G: Today is neither Tuesday nor Saturday. Given that among them only one statement is correct, which day of week is today? 有七個人在爭論今天是星期幾，他們的說法如下： A：昨天是星期三。 B：明天不是星期二。 C：明天是星期三。 D：後天是星期二。 E：今天是星期二。 F：今天不是星期一，也不是星期二，也不是星期天。 G：今天不是星期二或星期六。 已知七個人當中只有一個人說對了，那麼今天是星期幾？ Algebra 代數

6.

Given a is a real number, find the minimum value of a  2  a  3  a  4  a  5 . 已知 a 是一個實數，求 a  2  a  3  a  4  a  5 的最小值。

7.

Find the value of 13  12 13  12 ... . 求 13  12 13  12 ... 的值。

8.

9.

Given x is a real number, solve 4 x  (2 x 2 )  32  0 . x x 2 已知 x 為實數，求 4  (2 )  32  0 的解。  x  2 y 2  54  Given x and y are positive integers and  2 y  xy  30 , find the value of 2x  y .  x  2 y  14  x  2 y 2  54  已知 x 及 y 皆為正整數且  2 y  xy  30 ，求 2x  y 的值。  x  2 y  14

3 10. Given x is a non-zero real number and x 2  7 x  1  0 , find the value of x 

1 . x3

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

3 已知 x 為非零實數且 x 2  7 x  1  0 ，求 x 

1 的值。 x3

Number Theory 數論 2 11. Given x > 0, find the minimum value of 2 x  4 x  2 已知 x > 0，求 2 x  4 x 

16  11 . x2

16  11 的最小值。 x2

12. If 12  112  a  b , and both a and b are positive integers, find the value of a b . 若 12  112  a  b ，且 a、b 皆為正整數，求 a b 的值。

13. Find the value of x when

x 2  6 x  58  x 2  6 x  205 attains its minimum.

當 x 2  6 x  58  x 2  6 x  205 為最小時，求 x 的值。



x  9 (mod17) 14. Find the least positive integral solution of 2x  5 (mod 7) .



x  9 (mod17) 求 2x  5 (mod 7) 的最小正整數解。

15. If a、b、c are distinct prime numbers and ( a  b)( a  c)  75 , find the value of b + c. 已知 a、b、c 為相異質數且 ( a  b)( a  c)  75 ，求 b + c 的值。

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet Geometry 幾何 16. The figure shows a circle and AB, BC and CD are tangents to it. Tangents AB, BC and CD cut the circle at A, F and C respectively. If AB / / DC and AD  18 , find the value of BF  FC . 圖中 AB、BC、CD 是圓的切線且 AB // DC，且 A、D 和 F 分別是圓與 AB、CD 和 BC 的切點。若 AD = 18，求 BF  FC 的值。

Question 16 第 16 題 17. In the figure below, ABCD is a square. P is a point in it s.t. AP = a, BP = 2a and CP = 3a. Find the area of the square in terms of a. 在下圖中，ABCD 是一個正方形。P 是正方形內的一點使 AP = a、BP = 2a 及 CP = 3a。以 a 表示正 方形的面積。 A

D P

B

C

Question 17 第 17 題

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 18. In the figure, BC = 2015. Points D, E and F are on BC, CA and AB respectively such that DCEF is a parallelogram. If AFE  BFD , find the perimeter of DCEF. 在圖中，BC = 2015。D，E 和 F 分別在 BC，CA 和 AB 上使得 DCEF 為一平行四邊形。若 AFE  BFD ，求 DCEF 的周界。 A F

B

D

E C

Question 18 第 18 題  . Find CAM . 19. In the figure, AM  MC  。求 CAM 的值。 在下圖中， AM  MC

A

M

74

42

B

C Question 19 第 19 題

20. In the triangle ABC shown in the figure below, B  60 , C  45 and AC  6 . Find the area of ABC . 在下圖的 ABC 中， B  60 、 C  45 及 AC  6 。求 ABC 的面積。

A

B

C Question 20 第 20 題

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet Combinatorics 組合數學 21. Find the number of unlike terms of ( a  b  c) 20 after expansion. 20 求 ( a  b  c) 展開後的異類項數目。 22. 8 boys sit on their own chairs. Now they stand up and choose a chair to sit randomly. How many possible cases are there that “only one boy not sit on his own chair”? 8 個男孩原來坐在自己的位子上，現在他們站起來並隨機選擇一個椅子坐下。有多少個「只有一個 男孩沒有坐在自己的位子上」的情況？ 23. Given x1 , x2 , , x5 are all positive integers greater than 2, how many solutions are there to the inequality x1  x2    x5  20 ? 已知 x1 , x2 , , x5 皆為大於 2 的正整數，問不等式 x1  x2    x5  20 有多少個解組？ 24. Find the minimum number of positive integers to be chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 to ensure that there are 2 numbers such that one of them is a multiple of another. 從 1、2、3、4、5、6、7、8、9 與 10 中選取最少多少個正整數，使其中必有 2 個數字，某一個是另一個 的倍數？ 25. How many ways are there for 6 different balls to be put into 3 different boxes? (The boxes may be empty.) 把六個不同的球放進 3 個不同的盒子，有多少個方法？﹝盒子可以是空的﹞ ~ 全卷完 ~ ~ End of Paper ~

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

Solutions 題解 Logical Thinking 邏輯思維 1.

A box contains 100 coloured balls: 30 blue, 30 red, 20 green and 20 yellow. Sherry takes some balls from the box without looking at the colours of the balls. What is the least number of balls that she must take so that she has 25 balls with same colour? 有一箱顏色球共有 100 個：30 個藍色、30 個紅色、20 個綠色及 20 個黃色。雪莉需要在不看球的顏 色的情況下從箱中取出顏色球，那麼她必須最少取出多少個球才能讓她取得 25 個相同顏色的球？

解：89 To make sure 25 balls with same colours are drawn, we need to consider the worst possible outcome that all the green and yellow balls and also 24 blue balls and 24 red balls are drawn. So 20  20  24  24  1  89 pens should be drawn to make sure she draw 25 balls with same colour. 要肯定取得 25 個相同顏色的球，先考慮最不利的情況，即先取走所有黃色球及綠色球，且藍色球 和紅色球都要取得最少 24 個。 所以若她要取得 25 個相同顏色的球她需要取出 20  20  24  24  1  89 個球。

2.

According to the pattern shown below, what is the next number? 按以下規律，問下一個數是甚麼？ 2、9、28、65、126、217、344、…

解：513 Each number is the sum of a cubic number and 1. 每一項皆為立方數與 1 之和。 13  1  2 23  1  9 33  1  28 43  1  65 53  1  126 63  1  217 73  1  344 83  1  513

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

3.

1 2 . Find the integral part of 2 2 2     50 51 52 69 1 2 的整數部分。 求 2 2 2     50 51 52 69

解：1 1 1 1   2 2 2 2 2 2 2 2 2 2 2 2           50 51 52 69 50 51 52 69 69 69 69 69 1 1 1   40 2 2 2 2 40     50 50 51 52 69 69 1 1.25   1.725 2 2 2 2     50 51 52 69 1 2 is 1. The integral part of 2 2 2     50 51 52 69 1 2 2 2 2 的整數部份是 1。     50 51 52 69

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 4.

There are six teams named A, B, C, D, E and F, participating a tournament. In 5 days, each team will play one game in each day. They play another team once in the tournament. Given that: 1) Team A wins Team B on the first day. 2) Team C is defeated by Team D on the second day. 3) Team E wins Team A on the third day. 4) Team B wins Team C on the third day. Which team does Team B play with on the fifth day？ 有 A、B、C、D、E、F 六個籃球隊參加循環賽，賽程總共五天，每隊每天出賽一次，五天裡各與不同 的隊伍比賽一場，所以每天都有三場比賽，已知： 1) 第一天 A 隊贏了 B 隊 2) 第二天 C 隊輸給了 D 隊 3) 第三天 E 隊贏了 A 隊 4) 第四天 B 隊贏了 C 隊 請問，B 隊在第五天與哪一隊比賽？

解：Team F / F 隊 The arrangement of competition is shown in the following table. 各天賽事的安排如下表所示： Day Contest 1 Contest 2 比賽日 比賽一 比賽二

Contest 3 比賽三

1

A vs B

C vs E

D vs F

2

A vs F

B vs E

C vs D

3

A vs E

B vs D

C vs F

4

A vs D

B vs C

E vs F

5

A vs C

B vs F

D vs E

5.

There are 7 people arguing which day of week is today Their statements are quoted as below: A: Yesterday was Wednesday. B: Tomorrow will not be Tuesday. C: Tomorrow will be Wednesday. D: The day after tomorrow will be Tuesday. E: Today is Tuesday. F: Today is neither Monday, Tuesday nor Sunday. G: Today is neither Tuesday nor Saturday. Given that among them only one statement is correct, which day of week is today? 有七個人在爭論今天是星期幾，他們的說法如下： A：昨天是星期三。 B：明天不是星期二。 C：明天是星期三。 D：後天是星期二。 E：今天是星期二。 F：今天不是星期一，也不是星期二，也不是星期天。 G：今天不是星期二或星期六。 已知七個人當中只有一個人說對了，那麼今天是星期幾？ 解：Monday / 星期一 Converting their statements, we will have the following table: 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 經過轉換他們的說法可得出下表： A B C D E Monday 星期一 Tuesday 星期二 Wednesday 星期三 Thursday  星期四 Friday 星期五 Saturday 星期六 Sunday 星期日

F

G 



































Among them only Monday satisfies the condition that “only one statement is correct”. 當中只有星期一符合「只有一個人說對了」這個條件。

Algebra 代數 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

6.

Given a is a real number, find the minimum value of a  2  a  3  a  4  a  5 . 已知 a 是一個實數，求 a  2  a  3  a  4  a  5 的最小值。

解：4

a  2 a  2  a  3  a  4  a  5  4a  14  4(2)  14 6 3  a  2 a  2  a  3  a  4  a  5  a  2  a  3  a  4  a  5  2a  10  2(3)  10 4 4  a  3 a  2  a  3  a  4  a  5  a  2  a  3  a  4  a  5 4 5  a  4 a  2  a  3  a  4  a  5  a  2  a  3  a  4  a  5  2a  4  2(4)  4 4

a  5 a  2  a  3  a  4  a  5  a  2  a  3  a  4  a  5  4a  14  4(5)  14 6 a  2  a  3  a  4  a  5 is 4. The minimum value of 求 a  2  a  3  a  4  a  5 的最小值為 4。

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

7.

Find the value of 13  12 13  12 ... . 求 13  12 13  12 ... 的值。

解：1 Denote x  13  12 13  12 ... , 記 x  13  12 13  12 ... ， x 2  13  12 x x 2  12 x  13  0 ( x  13)( x  1)  0 x 1

8.

Given x is a real number, solve 4 x  (2 x 2 )  32  0 . x x 2 已知 x 為實數，求 4  (2 )  32  0 的解。

解：3 4 x  (2 x  2 )  32  0

 2   4  2   32  0  2  8  2  4   0 x 2

x

x

x

2x  8  0 2 x  23 x3

9.

 x  2 y 2  54  Given x and y are positive integers and  2 y  xy  30 , find the value of 2x  y .  x  2 y  14  x  2 y 2  54  已知 x 及 y 皆為正整數且  2 y  xy  30 ，求 2x  y 的值。  x  2 y  14

解：13  x  2 y 2  54  2 y 2  2 y  40   x  2 y  14 y 2  y  20  0 y5 y  4 (rejected) y  4 ﹝捨去﹞ 2 y  xy  30 x4 2 x  y  2  4  5  13 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

3 10. Given x is a non-zero real number and x 2  7 x  1  0 , find the value of x  3 已知 x 為非零實數且 x 2  7 x  1  0 ，求 x 

1 . x3

1 的值。 x3

解： 322 x2  7 x  1  0 x 2  1  7 x 1 x   7 x 3 1 x 3  3x   3  343 x x 1 x 3  3  343  3(7) x  322 Number Theory 數論 2 11. Given x > 0, find the minimum value of 2 x  4 x  2 已知 x > 0，求 2 x  4 x 

16  11 . x2

16  11 的最小值。 x2

解：15 2 x2  4x 

16  11 x2

 x2  4x  4  x2  8 

16  15 x2

4  ( x  2) 2  ( x  ) 2  15 x The expression attains its minimum when x  2 . 當 x  2 時，該算式得最小值。

12. If 12  112  a  b , and both a and b are positive integers, find the value of a b . 若 12  112  a  b ，且 a、b 皆為正整數，求 a b 的值。

解： 2 7 12  112  a  b 12  2 28  a 2  2a b  b a b  28 2 7 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

13. Find the value of x when

x 2  6 x  58  x 2  6 x  205 attains its minimum.

當 x 2  6 x  58  x 2  6 x  205 為最小時，求 x 的值。 解： 1 x 2  6 x  58  x 2  6 x  205  ( x  3) 2  7 2  ( x  3) 2  142 Consider the value of ( x  3) 2  7 2  ( x  3) 2  142 is equal to the sum of distance between (3 , 7), (x , 0) and ( 3 , 14 ), (x , 0), the minimum value attains when (3 , 7), (x , 0) and ( 3 , 14 ) lie on a 7  14 7  0  straight line. Hence , x 1. 33 3 x ( x  3) 2  72  ( x  3) 2  142 的最小值即求(3 , 7), (x , 0) 與( 3 , 14 ), (x , 0)距離之和的最小值。 考慮(3 , 7)、(x , 0)及 ( 3 , 14 ) 成一直線，故得

14. Find the least positive integral solution of





7  14 7  0  ， x 1。 33 3 x

x  9 (mod17) . 2x  5 (mod 7)

x  9 (mod17) 求 2x  5 (mod 7) 的最小正整數解。 解：111 x  9 (mod17) x  9 (mod17)  2x  5 (mod 7) x  6 (mod 7) x  9 (mod17)  x  17 p  9 17 p  9  6 (mod 7) 3 p  9  6 (mod 7) p  6 (mod 7) p  7k  6 x  17(7k  6)  9  119k  111 The least positive integral solution of x is 111. x 的最小正整數解為 111。





請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 15. If a、b、c are distinct prime numbers and ( a  b)( a  c)  75 , find the value of b + c. 已知 a、b、c 為相異質數且 ( a  b)( a  c)  75 ，求 b + c 的值。 解：24 As a  b and a  c are odd numbers, and the difference between two odd numbers should be even, we have a  2 . Without generosity, let b  c . (2  b)(2  c)  75 (b  2)(c  2)  52  3 (b  2, c  2)  (1, 75) / (3, 25) / (5, 15) (b, c)  (3, 77) / (5, 27) / (7,17) Only a number pair (7,17) satisfies the requirement, hence b  c  24 . a  b 和 a  c 均為奇數。因兩奇數相減為偶數，故 a  2 。 不失一般性，設 b  c 。 (2  b)(2  c)  75 (b  2)(c  2)  52  3 (b  2, c  2)  (1, 75) / (3, 25) / (5, 15) (b, c)  (3, 77) / (5, 27) / (7,17) 只有數組 (7,17) 符合要求，可得 b  c  24 。

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet Geometry 幾何 16. The figure shows a circle and AB, BC and CD are tangents to it. Tangents AB, BC and CD cut the circle at A, F and C respectively. If AB / / DC and AD  18 , find the value of BF  FC . 圖中 AB、BC、CD 是圓的切線且 AB // DC，且 A、D 和 F 分別是圓與 AB、CD 和 BC 的切點。若 AD = 18，求 BF  FC 的值。

Question 16 第 16 題 解：81 FBO  FCO  180  2  90 BFO ~ OFC BF OF  OF FC BF  FC  OF 2 AD 2 ( ) 2 =81

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 17. In the figure below, ABCD is a square. P is a point in it s.t. AP = a, BP = 2a and CP = 3a. Find the area of the square in terms of a. 在下圖中，ABCD 是一個正方形。P 是正方形內的一點使 AP = a、BP = 2a 及 CP = 3a。以 a 表示正 方形的面積。 A

D P

B

C

Question 17 第 17 題 解： (5  2 2)a 2 Rotate ABP to CBP ' around B for 90 clockwise as in the figure below. Then BP '  BP  2a . CP '  AP  a . Connect PP ' . PBP '  PBC  CBP '  PBC  ABP  90 Therefore PP '  PB 2  P ' B 2  (2a) 2  (2a ) 2  2 2a . Since PP '2  P ' C 2  8a 2  a 2  9a 2  PC 2 , PP ' C  90 . APB  CP ' B  BP ' P  PBP '  90  45  135  BC 2 2 2 Area of the square  a  4a  2(a)(2a) cos135  (5  2 2)a 2 沿 B 點順時針旋轉 ABP 90 至 CBP ' ，如圖所示。 可得 BP '  BP  2a 、 CP '  AP  a 。 連接 PP ' ，得 PBP '  PBC  CBP '  PBC  ABP  90 。 由此 PP '  PB 2  P ' B 2  (2a) 2  (2a ) 2  2 2a 。 由於 PP '2  P ' C 2  8a 2  a 2  9a 2  PC 2 ，得 PP ' C  90 。 APB  CP ' B  BP ' P  PBP '  90  45  135 。  BC 2  a 2  4a 2  2(a)(2a ) cos135 正方形面積  (5  2 2)a 2 D

A P

B

C

P' 18. In the figure, BC = 2015. Points D, E and F are on BC, CA and AB respectively such that DCEF is a 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet parallelogram. If AFE  BFD , find the perimeter of DCEF. 在圖中，BC = 2015。D，E 和 F 分別在 BC，CA 和 AB 上使得 DCEF 為一平行四邊形。若 AFE  BFD ，求 DCEF 的周界。 A F

B

D

E C

Question 18 第 18 題 解：4030 FE / / BC and FD / / AC , ABC  AFE  BFD  BAC ABC , AFE and FBD are all isosceles triangles and hence CE  DF  BD . Perimeter of DCEF  2(CE  DC )  2( BD  DC )  2BC  4030 .  FE / / BC 和 FD / / AC ABC  AFE  BFD  BAC ABC ， AFE 和 FBD 皆等腰，而 CE  DF  BD 。 DCEF 的周界  2(CE  DC )  2( BD  DC )  2BC  4030 。  . Find CAM . 19. In the figure, AM  MC  。求 CAM 的值。 在下圖中， AM  MC

A

74 B

M

42 C

Question 19 第 19 題 解： 37 CAM 1  ABC 2 1 CAM   74 2  37 20. In the triangle ABC shown in the figure below, B  60 , C  45 and AC  6 . Find the area of ABC . 請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 在下圖的 ABC 中， B  60 、 C  45 及 AC  6 。求 ABC 的面積。

A

B

C Question 20 第 20 題

3 3 2 Let h be the length of the height of ABC corresponding to BC. h2  h2  6 We have h 3

解：

3 3  tan 60 tan 45 BC  1  3 BC 

3(1  3) 2 The area of ABC 3 3  2 設h為三角形 ABC 對應BC的高。 h2  h2  6 可得 ， h 3 

3 3  tan 60 tan 45 BC  1  3 BC 

3(1  3) 2 ABC 的面積 3 3  2 

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet Combinatorics 組合數學 21. Find the number of unlike terms of ( a  b  c) 20 after expansion. 20 求 ( a  b  c) 展開後的異類項數目。 解：231 Every term in the expansion of ( a  b  c) 20 is in the form of a n1 b n2 c n3 where n1 , n2 , n3 are all nonnegative and n1  n2  n3  20 . 22 Number of unlike terms is C2  231 ( a  b  c) 20 展開後的每一項皆可以 a n1 b n2 c n3 表示，且 n1 , n2 , n3 為非負整數及 n1  n2  n3  20 。 22 由此異類項數目為 C2  231 。

22. 8 boys sit on their own chairs. Now they stand up and choose a chair to sit randomly. How many possible cases are there that “only one boy not sit on his own chair”? 8 個男孩原來坐在自己的位子上，現在他們站起來並隨機選擇一個椅子坐下。有多少個「只有一個 男孩沒有坐在自己的位子上」的情況？ 解：0 If one of them does not sit on his own chair, there should be another student not sitting on his chair. So there does not exist a case that only one boy does not sit on his own chair. 由於若有一名男孩沒有坐在自己的位子，必然有另一位男孩同樣沒坐在自己的位子，所以不存在 只有一個男孩沒坐在自己的位子的情況。

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 23. Given x1 , x2 , , x5 are all positive integers greater than 2, how many solutions are there to the inequality x1  x2    x5  20 ? 已知 x1 , x2 , , x5 皆為大於 2 的正整數，問不等式 x1  x2    x5  20 有多少個解組？ 解：252

a1  x1  2 a2  x2  2 Denote a3  x3  2 , a4  x4  2 a5  x5  2 we have x1  x2    x5  20  a1  a2    a5  10 and x1  x2    x5  20  a1  a2    a5  10 9 Number of solutions for equation a1  a2    a5  10 is C4  126 . Inequality x1  x2    x5  20 is equivalent to the equation a1  a2    a5  a6  10 where a6 is also a 9 positive integer. So the number of solutions is C5  126 . As a result there are 126  126  252 solutions. a1  x1  2 a2  x2  2 記 a3  x3  2 ， a4  x4  2 a5  x5  2 可得等式 x1  x2    x5  20  a1  a2    a5  10 及 x1  x2    x5  20  a1  a2    a5  10 。 9 等式 a1  a2    a5  10 的解組數目為 C4  126 。 不等式 a1  a2    a5  20 等價於等式 a1  a2    a5  a6  10 且 a6 是正整數。所以解組數目為

C59  126 。 綜合兩個情況，可得解組數目為 126  126  252 。

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet 24. Find the minimum number of positive integers to be chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 to ensure that there are 2 numbers such that one of them is a multiple of another. 從 1、2、3、4、5、6、7、8、9 與 10 中選取最少多少個正整數，使其中必有 2 個數字，某一個是另一個 的倍數？ 解：6 Separate the numbers into 5 groups: {1 , 2 , 4 , 8} {3 , 6} {5 , 10} {7} {9} If any 2 numbers are in the same group, then one of them is a multiple of another. By the pigeonhole principle, if 6 numbers are chosen from them, then there are 2 of them in the same group. 把 1 至 10 分成 5 組： {1 , 2 , 4 , 8} {3 , 6} {5 , 10} {7} {9} 任何兩個在同一組的數字，他們與另一個都是倍數關係。 根據抽屜原理，若從中選取 6 個數字，則一定有兩個數字在同一組中。 因此，必定有其中 2 個數字，某一個是另一個的倍數。 25. How many ways are there for 6 different balls to be put into 3 different boxes? (The boxes may be empty.) 把六個不同的球放進 3 個不同的盒子，有多少個方法？﹝盒子可以是空的﹞ 解：729 For every ball it has 3 choices, so the number of ways for 6 balls to be put into 3 different boxes is 36  729 . 對每個球都有 3 個盒子選擇，所以把 6 個球放到 3 個盒子則有 36  729 個方法。

~ 全卷完 ~ ~ End of Paper ~

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.

泰國國際數學競賽 2015 中學三年級 試題集 Thailand International Mathematical Olympiad 2015 Secondary 3 Past Paper Booklet

泰國國際數學競賽 2015 THAILAND INTERNATIONAL MATHEMATICAL OLYMPIAD 2015

中學三年級 答案 Secondary 3 Answer Key Question No 題號

Answer 答案

Question No 題號

Answer 答案

Question No 題號

Answer 答案

1

89

11

15

21

231

2

513

12

2 7

22

0

3

1

13

1

23

252

4

Team F / F 隊

14

111

24

6

5

Monday / 星期一

15

24

25

729

6

4

16

81

7

1

17

(5  2 2)a 2

8

3

18

4,030

9

13

19

37

10

322

20

3 3 2

請以最簡形式填寫答案，若計算結果是分數，請確保為真分數或帶分數，或將計算結果寫成小數。錯誤單位將不給予任何分數。 Write down the answer in the simplest form. If the calculation result is a fraction, please write down the answer as a proper or mixed fraction, decimal figure is also accepted. Marks will NOT be given for incorrect unit.