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泰國國際數學競賽模擬試 2022 - 2023

THAILAND INTERNATIONAL
MATHEMATICAL OLYMPIAD
MOCK EXAM 2022 - 2023

中學三年級 Secondary 3
時限：90 分鐘
Time allowed: 90 minutes
模擬試題
Mock Paper
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 6 題，共 30 題，每題 5 分，總分 150 分，答錯不扣分。
There are 5 exam areas and 6 questions in each exam area. There are a total of 30 questions in this
Question-Answer Book. Each carries 5 marks. Total score is 150 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.

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

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.

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

填空題（第 1 至 30 題）（每題 5 分，答錯及空題不扣分）

Open-Ended Questions (1st ~30th) (5 points for correct answer, no penalty point for wrong answer)

Logical Thinking
邏輯思維

1. The sum of 9 consecutive odd numbers is 207. Find the third largest number.
某 9 個連續奇數之和是 207，求第三大的數字。

2. There are 13 questions in a mathematics competition. The scores of each question are allocated in the
following ways: 1 mark will be given for a correct answer, 1 mark will be deducted from a wrong answer
and 0 marks will be given for a blank answer. Find the minimum number of candidate(s) participating in
the competition to ensure that at least 4 of the candidates will have the same scores in the competition.
某數學比賽共有 13 條題目。以下述方式為每個題目評分：答對得 1 分、答錯倒扣 1 分、不作答
得零分。求最小參加此比賽的參賽者的數目才可保證比賽中至少有 4 人同分。

3. 16th June, 2022 is Thursday, which day of the week will 16th June, 2029 be?
2022 年 6 月 16 日是星期四，那麼 2029 年的 6 月 16 日會是星期幾？

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

4. Given that A : B = 3 : 4 , B : C = 2 :1 , C : D = 5 : 9 , D : E = 3 :1 and A + B + C + D + E = 20700 . What

is the value of D?
已知 A : B = 3 : 4 、 B : C = 2 :1 、 C : D = 5 : 9 、 D : E = 3 :1 ，且 A + B + C + D + E = 20700 ，問 D 是多
少？

5. Use decimal number system to represent quaternary system 20234 .

請以十進制表示四進制數 20234 。

6. According to the pattern shown below, what is the number in the blank?
按以下規律，求在橫線上的數字。
1 、 1 、 2 、 7 、 34 、 203 、 1420 、 __ 、…

Algebra
代數

7. Simplify 20 + 2 91 .
化簡 20 + 2 91 。

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

8. If 2 x 4 − 9 x 3 + mx 2 + nx + 54 = 0 is divisible by x 2 + 4 x + 3 , find m+n.

若 2 x 4 − 9 x 3 + mx 2 + nx + 54 = 0 能被 x 2 + 4 x + 3 整除，求 m+n。

9. Factorize 4 x 2 − 4 y 2 + 40 x − 28 y + 51 .
因式分解 4 x 2 − 4 y 2 + 40 x − 28 y + 51 。

10. It is known that x is rational and x = 72 − 6 72 − 6 . Find the value of x.

已知有理數 x = 72 − 6 72 − 6 ，求 x 的值。

11. Given that x is a real number, find the maximum value of − x 2 − 10 x − 91 .

已知 x 為實數，求 − x 2 − 10 x − 91 的最大值。

1 1
12. Given x  0 and x + = 11 , find the value of x 2 + 2 .
x x
1 1
已知 x  0 及 x + = 11 ，求 x 2 + 2 的值。
x x

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

Number Theory
數論

13. Find the sum of all positive factors of 756.

求 756 的所有正因數之和。

21
14. It is known that x is rational, x  0 and x = 4 + . Find the value of x.
21
4+
4+
21
已知有理數 x  0 且 x = 4 + ，求 x 的值。
21
4+
4+

15. Given that 239 A85367 B is a 10-digit number which is divisible by 7, find the sum of all the possible
value of A + B .
已知 239 A85367 B 是一個十位數，且可被 7 整除，求 A + B 的所有可能值的總和。

16. If x  0 , x is an odd 3-digit number and x  17 ( mod 24 ) , find the minimum value of x.
若 x  0 且 x  17 ( mod 24 ) ，求 x 的最小三位奇數值。

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

17. Given a and b are real numbers and satisfy equations a − b = 9 and ab = 13 , find the value of a 2 + b 2 .
已知 a、b 皆為實數且滿足等式 a − b = 9 和 ab = 13 ，求 a 2 + b 2 的值。

18. Find the remainder when 4912 is divided by 8.

求當 4912 除以 8 時所得的餘數。

Geometry
幾何

19. The side lengths of a triangle are 7, 12 and 17. Find the area of this triangle. (Answer in terms of simplest
surd form)
某三角形的邊長分別為 7、12 與 17，求三角形的面積。（答案以最簡根式表示）

sin B
20. In acute ABC , AB = 17 , BC = 18 and AC = 6 , find the value of 2 .
sin A
sin B
在銳角 ABC ， AB = 17 ， BC = 18 和 AC = 6 。求 2 的值。
sin A

21. Find the shortest distance from the point ( 9,5 ) to y − 2 x + 5 = 0 . (Answer in terms of simplest surd
form)
求點 ( 9,5 ) 與 y − 2 x + 5 = 0 的最短距離。（答案以最簡根式表示）

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

22. Find the area enclosed by the y-axis the straight lines 9 x + 3 y − 27 = 0 and y = 2 .
求由 y-軸、直線 9 x + 3 y − 27 = 0 及 y = 2 圍出的面積。

23. M ( x, 7 ) is the mid-point of A (1, − y ) and B ( 35, −15 ) . Find the value of x − y .
M ( x, 7 ) 是 A (1, − y ) 和 B ( 35, −15 ) 的中點，求 x − y 的值。

24. If a straight line L passes through A ( −6,5 ) , and the slope of L is −2 . Find the y-intercept of L.
若直線 L 通過 A ( −6,5 ) ，而 L 的斜率為 −2 ，求 L 的 y 截距。

Combinatorics
組合數學

25. 39 cards are marked from 1 to 39 and 2 cards are drawn at random without replacement. Find the
probability that one card drawn is a multiple of 3 and another one is a multiple of 5.
由 1 至 39 號卡片中不重複的隨意抽出 2 張，求得到一張號碼為 3 的倍數且另一張號碼為 5 的倍
數的卡片概率。

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

26. Julia draws 5 hats randomly from a bag containing 6 red hats and 5 yellow hats. Find the probability that
at most 3 yellow hats are drawn.
茱莉亞從一個裝有 6 頂紅色帽子及 5 頂黃色帽子的袋子隨機抽出 5 頂帽子。求抽出最多 3 頂黃
色帽子的概率。

27. If x and y are positive integers such that 13 x + 3 y = 239 . Find the maximum value of xy .
已知 x、y 為正整數且 13 x + 3 y = 239 ，求 xy 的最大值。

28. It is known that a, b, c, d are integers and all of them are greater than 5. Find the number of solution sets
of a + b + c + d = 29 .
已知 a, b, c, d 為大於 5 的整數，求方程 a + b + c + d = 29 的解組數目。

29. A fair 6-face die is thrown 3 times. Find the probability that the product of numbers obtained is 24.
擲一枚均質六面骰子三次。求擲得點數之積為 24 的概率。

30. Find the number of 3-digit number(s) such that the hundreds digit is smaller than the tens digit and the
tens digit is smaller than the unit digit.
求三位數的數量使得它的百位值小於十位值，而十位值小於個位值。

~ 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.