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泰國國際數學競賽總決賽 2017 - 2018 Thailand International Mathematical Olympiad Final Round 2017 - 2018 Primary 3 小學三年級





Question Paper 試題

填空題（第 （每題 填空題（第 1 1至 至 30 30 題） 題） （每題 5 5 分，答錯及空題不扣分） 分，答錯及空題不扣分） st th st ~30th) (5 points for correct answer, no penalty point for Open-Ended Questions (1 Open-Ended Questions (1 ~30 ) (5 points for correct answer, no penalty point for wrong wrong answer) answer)

Logical Thinking 邏輯思維 1. 1.

2. 2.

3. 3.

What What is is the the value value of of the the number number to to represent represent “?” “?” in in the the following following table? table? 求在下表中？代表的值。 求在下表中？代表的值。 11

22

33

44

55

3 3

4 4

55

66

77

4 4

9 9

？ ？

25 25

36 36

st According Accordingthto to the the pattern pattern shown shown below, below, how how many many squares squares are are there there from from the the 1 1st to to the the 134 134th symbol symbol counting counting from from the the left? left? 按以下規律，由左數起第 1 至第 按以下規律，由左數起第 1 至第 134 134 個符號中共有多少個正方形？ 個符號中共有多少個正方形？

○ ○△ △□ □△ △○ ○○ ○△ △□ □△ △○ ○ ○… ○…

The The age age of of Samuel Samuel 7 7 years years later later is is equal equal to to 2 2 times times of of the the age age of of Joseph Joseph 5 5 years years ago. Given Samuel is 19 years old now, how old is Joseph now? ago. Given Samuel is 19 years old now, how old is Joseph now? 森美 森美 77 年後的年齡是約瑟夫 年後的年齡是約瑟夫 5 5 年前的年齡的兩倍。已知森美今年 年前的年齡的兩倍。已知森美今年 19 19 歲，那 歲，那 麼約瑟夫今年幾多歲？ 麼約瑟夫今年幾多歲？

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Final Round

4.

John wrote a 3-digit number on a piece of paper and asked Peter to guess it. Peter asked: “Is the number 124?” John replied: “No, but the number is a multiple of 124.” Peter asked again: “Is the number divisible by 6?” John replied: “No, you will get a remainder of 2 when the number divided by 6.” Peter asked again: “Is the number divisible by 8?” John said: “No.” Given all the digits in that 3-digit number are all different, what is the number written by John? 約翰寫一個 3 位數在一張紙上，吩咐彼得來猜猜它。 彼得問：「這個數是否 124？」 約翰回覆： 「不是，但這個數是 124 的倍數。」 彼得再問： 「這個數是否能被 6 整除？」 約翰回覆： 「不是，這個數除以 6 餘 2。」 彼得再問： 「這個數是否能被 8 整除？」 約翰回覆： 「不是。」 已知這個三位數的三個數位都不一樣，求約翰寫出來的數字。

5.

According to the pattern shown below, how many  are there in the 9th group?按 以下規律，問第� 9 組有多少個 ？�



1st Group

第 1 組











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

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





2nd Group 第 2 組

3rd Group 第 3 組

4th Group 第4組

Question 5 第5題 36

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6. 6.

Now it gets dark so mum asked Amy to light up. The lamp is on now but Amy Now it gets dark so mum asked Amy to light up. The lamp is on now but Amy wants to play trick on mum and switch on the lamp 39 times. Is the lamp on/ off wants to play trick on mum and switch on the lamp 39 times. Is the lamp on/ off now? (Please answer on / off as your final answer) now? (Please answer on / off as your final answer) 天已經黑了，現時燈已經開啟了，艾美想捉弄一下媽媽，連按了 39 次燈， 天已經黑了，現時燈已經開啟了，艾美想捉弄一下媽媽，連按了 39 次燈， 現時燈是亮，還是不亮？（請以亮或不亮作最後答案） 現時燈是亮，還是不亮？（請以亮或不亮作最後答案）

Arithmetic Arithmetic Arithmetic 算術 算術 算術 7. 7.

Find the value of 222  74  788  316 . Find the 74 value of 316的值。 74  788  316 . 求 222  788 222 求 222  74  788  316 的值。

8. 8.

Find the value of 34 11  3313  11 27 . Find the value of 34 11  3313  11 27 . 求 34 11  3313  11 27 的值。 求 34 11  3313  11 27 的值。

9. 9.

Find the value of 11111111 . Find the value of 11111111 . 求 11111111 的值。 求 11111111 的值。

10. Find the value of 11  22  33  44  55  66  77  88  99 . 10. 求 Find the value of    33  77  88  99 . 的值。 11  22  33  4411 5522  66  7744  8855  9966 求 11  22  33  44  55  66  77  88  99 的值。 11. Find the value of 4 16  25  625 . 11. Find the value of 4 16  25  625 . 求 4 16  25  625 的值。 求 4 16  25  625 的值。 12. Find the value of 2  4  6  8  10 12  14 16  18  20  22  24  26 . 12. 求 Find the value of 6 816  10 12  14 16  18  22  24  26 . 的值。 2  4  6  8  10212414  18  20  22  24  2620 求 2  4  6  8  10 12  14 16  18  20  22  24  26 的值。

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13. 13. 13. 13. 13. 13. 14. 14. 14. 14. 14. 14.

Final Round

15. 15. 15. 15. 15. 15.

16. 16. 16. 16. 16. 16. 17. 17. 17. 17. 17. 17. 18. 18. 18. 18. 18. 18.

38

Number Theory Number Theory 數論 Number Theory Number Theory 數論 Number Theory 數論 數論 Number Theory Number Theory Define the operation symbol a  b 數論  a  b  b  6 , find the value of (17 18) . Define the operation symbol a  b 數論 數論 a  b  b  6 , find the value of (17 18) . Define theaoperation 定義運算 的值。 a  b(17  a 18) bb  6 , find the value of (17 18) .  b  a symbol b  b  6 ，求 Define theaoperation a  b(17  a 18) bb  6 , find the value of (17 18) . 定義運算 的值。  b  a symbol b  b  6 ，求 Define the operation symbol a  b  a  b  b  6 , find the value of (17 18) . 定義運算 a  b  a  b  b  6 ，求 (17 18) 的值。 Define theaoperation 定義運算 的值。 a  b(17  a 18) bb  6 , find the value of (17 18) .  b  a symbol b  b  6 ，求 定義運算 a  b  a  b  b  6 ，求 (17 18) 的值。 定義運算 a  b  a  b  b  6 ，求 (17 18) 的值。 The difference of A and B is 201. The sum between A and B is 823. Given B is The difference of A and B is 201. The sum between A and B is 823. Given B is The of A B isof201. largerdifference than A, find theand value A. The sum between A and B is 823. Given B is The of A B isof201. largerdifference than A, find theand value A. The sum between A and B is 823. Given B is The of A and B isof201. The sum between A and 823. Given B is A、B 兩數之差是 201，A、B 兩數之和是 823。已知 B 比BAis大，求 A 的值。 largerdifference than A, find the value A. The of A and B isof201. The sum between A and 823. Given B is largerdifference than A, find the value A. A、B 兩數之差是 201，A、B 兩數之和是 823。已知 B 比BAis大，求 A 的值。 larger 兩數之差是 than A, find the value of A. A、B 201，A、B 兩數之和是 823。已知 B 比 A 大，求 A 的值。 A、B 201，A、B 兩數之和是 823。已知 B 比 A 大，求 A 的值。 larger 兩數之差是 than A, find the value of A. A、B 兩數之差是 201，A、B 兩數之和是 823。已知 B 比 A 大，求 A 的值。 A、B 兩數之差是 201，A、B 兩數之和是 823。已知 B 比 A 大，求 A 的值。 The numbers below follow the arithmetic sequence, what is the sum of the 8th The numbers below follow the arithmetic sequence, what is the sum of the 8th rd The numbers below follow the arithmetic sequence, what is the sum of the 8th term and the 23 rd term? The numbers below follow the arithmetic sequence, what is the sum of the 8th term and the 23 term? rd The numbers below follow the arithmetic sequence, what is the sum of the 8th 根據以下的等差數列，問第 8 項及第 23 項之和是甚麼？ term and the 23 term? rd The numbers below follow the arithmetic sequence, what is the sum of the 8th term and the 23 根據以下的等差數列，問第 8 項及第 23 項之和是甚麼？ rd term? term and the 23rd term? 根據以下的等差數列，問第 項及第 2350, 項之和是甚麼？ 18,826, 34, 42, … 根據以下的等差數列，問第 項及第 2350, 項之和是甚麼？ term and the 23 term? 18,826, 34, 42, … 根據以下的等差數列，問第 項及第 2350, 項之和是甚麼？ 18,826, 34, 42, … 根據以下的等差數列，問第 項及第 2350, 項之和是甚麼？ 18,826, 34, 42, … 18, 26, 34, 42, 50, … 18, 26, 34, 42, 50, … The sum of A and B is 333. B is 8 times of A. Find the value of B. The sum of A and B is 333. B is 8 times of A. Find the value of B. A、Bsum 兩數之和是 8 倍。求 B 的值。 The of A and B333，B is 333. 是 B isA8的 times of A. Find the value of B. The of A and B333，B is 333. 是 B isA8的 times of A. Find the value of B. A、Bsum 兩數之和是 8 倍。求 B 的值。 The sum of A and B333，B is 333. 是 B isA8的 times of A. Find the value of B. A、B 兩數之和是 8 倍。求 B 的值。 A、B 兩數之和是 8 倍。求 B 的值。 The sum of A and B333，B is 333. 是 B isA8的 times of A. Find the value of B. A、B 兩數之和是 333，B 是 A 的 8 倍。求 B 的值。 A、B 兩數之和是 333，B 是 A 的 8 倍。求 B 的值。 What is the largest 3-digit number that can be divisible by 3 and 7? What is the largest 3-digit number that can be divisible by 3 and 7? 求最大的三位數能同時被 3 及 7 that 整除。 What is the largest 3-digit number can be divisible by 3 and 7? What is the largest 3-digit number can be divisible by 3 and 7? 求最大的三位數能同時被 3 及 7 that 整除。 What is the largest 3-digit number can be divisible by 3 and 7? 求最大的三位數能同時被 3 及 7 that 整除。 求最大的三位數能同時被 3 及 7 that 整除。 What is the largest 3-digit number can be divisible by 3 and 7? 求最大的三位數能同時被 3 及 7 整除。 求最大的三位數能同時被 3 及 7 整除。 The process 「  」has the following results： The process 「  」has the following results： The process 「  」has the following results： 已知程序計算「 」有以下結果： The process 「  」has the following results： 」有以下結果： 已知程序計算「 The process 「  」has the following 已知程序計算「 」有以下結果： 5 1  5 results： The process 「  」has the following 」有以下結果： 已知程序計算「 5 1  5 results： 已知程序計算「  」有以下結果： 54  3  12 54  12 已知程序計算「  」有以下結果： 54  12 54  3  12 5 7 574 63 512 210 4  132 4  132 7 574 63 512 210 5 Find the value of (10  4) . 74  32  74 63 512 210 Find the value of (10  4) . 74  32  74 63 512 210 Find value of (10  4) . 7  3  7  6  5  210  4) 求 (10the 的值。 Find value of (10  4) . 7  3  7  6  5  210  4) 求 (10the 的值。 Find value of (10  4) .  4) 求 (10the 的值。 Find value of (10  4) .  4) 求 (10the 的值。 求 (10  4) 的值。 求 (10  4) 的值。 奧冠教育出版社 Olympiad Champion Education Publisher 版權所有 不得翻印 All Rights Reserved

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Geometry 幾何 19. How many square(s) is / are there in the figure below? 請問下圖有多少個正方形？

Question 19 第 19 題 20. A prism has 2018 vertices, how many edge(s) does this prism have? 有一個柱體有 2018 個頂點，問這個柱體有多少條稜？ 21. At most how many acute angle(s) could a pentagon (5-sided polygon) contain? 一個五邊形最多可以有多少個銳角？ 22. At least how many squares can be seen if observing the figure below from the top? 如果從上方觀察下圖的立體，最少可以看見多少個正方形？

Question 22 第 22 題 23. At most how many piece(s) can be formed by cutting 5 times on a cake? 在蛋糕上切 5 下，最多可分成多少份？

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24. There are two squares in the figure below. Find the perimeter of the figure. 附圖中有 2 個正方形，求附圖的周界。 24. There are two squares in the figure below. Find the perimeter of the figure. 附圖中有 2 個正方形，求附圖的周界。 18 12 18

Question 24 第 24 題

12

Final Round

Combinatorics

25. After Amy gives 17 apples to Andy and get 8 apples from Johnny, they have 組合數學 equal number of apples. How many apples more did Johnny have than Andy originally? 25. After Amy gives 17 apples to Andy and get 8 apples from Johnny, they have 當艾美把 17 個蘋果送給安迪及取了強尼 8 個蘋果後，他們擁有的蘋果數目 equal number of apples. How many apples more did Johnny have than Andy 一樣。問原本強尼比安迪多多少個蘋果？ originally? 當艾美把 17 個蘋果送給安迪及取了強尼 8 個蘋果後，他們擁有的蘋果數目 一樣。問原本強尼比安迪多多少個蘋果？ 26. Choose 3 digits, without repetition, from 1, 2, 4, 6 to construct 3-digit numbers. Of these 3-digit numbers, how many of them are odd numbers? 從 1、2、4、6 中選 3 個不可重複的數位組成三位數。這些三位數當中有 26. Choose 3 digits, without repetition, from 1, 2, 4, 6 to construct 3-digit numbers. 多少個是奇數？ Of these 3-digit numbers, how many of them are odd numbers? 從 1、2、4、6 中選 3 個不可重複的數位組成三位數。這些三位數當中有 多少個是奇數？ 27. Numbers are drawn from the 35 integers 1 to 35. At least how many numbers are drawn at random to ensure that there are two numbers whose sum is 20? 在 1 至 35 這 35 個整數中最少任意選出多少個，才必定有兩個數之和是 20？ 27. Numbers are drawn from the 35 integers 1 to 35. At least how many numbers are drawn at random to ensure that there are two numbers whose sum is 20? 在 1 至 35 這 35 個整數中最少任意選出多少個，才必定有兩個數之和是 20？ 28. What is the smallest 4-digit number which is multiple of 5 and 7 and no digits are repeated? 求 5 和 7 的倍數且沒有數位重複最小 4 位數的值。 28. What is the smallest 4-digit number which is multiple of 5 and 7 and no digits are repeated? 求 5 和 7 的倍數且沒有數位重複最小 4 位數的值。 40

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29. If Andy goes from point B to point A, each step can only move up or move right. How many ways are there? 如果安迪從 B 點走到 A 點，每步只能向上或向右走，共有多少種走法？ A

B

Question 29 第 29 題 30. A teacher is distributing some candies among students. If the candies are distributed evenly among 5 people, 2 candies remain; if the candies are distributed evenly among 7 people, 4 candies remain; if the candies are distributed evenly among 9 people, 4 candies remain. How many candies does the teacher have at least? 老師有若干顆糖果，如果平分給 5 人，餘 2 顆 ；如果平分給 7 人，餘 4 顆；如果平分給 9 人，餘 4 顆。問老師最少有多少顆糖果？

~ End of Paper ~ ~ 全卷完 ~

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泰國國際數學競賽總決賽 2017 - 2018 Thailand International Mathematical Olympiad Final Round 2017 - 2018 Primary 3 小學三年級





Detailed Solution 詳細解答

Logical Thinking 邏輯思維 1.

What is the value of the number to represent “?” in the following table? 求在下表中？代表的值。 1

2

3

4

5

3

4

5

6

7

4

9

？

25

36

Answer: 16 2 4 1  9

4  6  1 25 3 5 1  16 2.

According to the pattern shown below, how many squares are there from the 1st to the 134th symbol counting from the left? 按以下規律，由左數起第 1 至第 134 個符號中共有多少個正方形？

Answer: 27

○ △ □ △ ○ ○ △ □ △ ○ ○…

○ △ □ △ ○). 規律為（○ △ □ △ ○）。 The pattern is (

134  5  26...4 26  1 27

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3.

The age of Samuel 7 years later is equal to 2 times of the age of Joseph 5 years ago. Given Samuel is 19 years old now, how old is Joseph now? 森美 7 年後的年齡是約瑟夫 5 年前的年齡的兩倍。已知森美今年 19 歲，那 麼約瑟夫今年幾多歲？

Answer: 18 (19  7)  2  5  13  5  18 4.

Final Round

John wrote a 3-digit number on a piece of paper and asked Peter to guess it. Peter asked: “Is the number 124?” John replied: “No, but the number is a multiple of 124.” Peter asked again: “Is the number divisible by 6?” John replied: “No, you will get a remainder of 2 when the number divided by 6.” Peter asked again: “Is the number divisible by 8?” John said: “No.” Given all the digits in that 3-digit number are all different, what is the number written by John? 約翰寫一個 3 位數在一張紙上，吩咐彼得來猜猜它。 彼得問：「這個數是否 124？」 約翰回覆： 「不是，但這個數是 124 的倍數。」 彼得再問： 「這個數是否能被 6 整除？」 約翰回覆： 「不是，這個數除以 6 餘 2。」 彼得再問： 「這個數是否能被 8 整除？」 約翰回覆： 「不是。」 已知這個三位數的三個數位都不一樣，求約翰寫出來的數字。

Answer: 620

620  124  5 620  6  103...2 620 4 155

132

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According to the pattern shown below, how many  are there in the 9th group? 按以下規律，問第 9 組有多少個 ？

5.



























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1st Group 第 1 組



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2nd Group 第 2 組

3rd Group 第 3 組

4th Group 第4組

Question 5 第5題 Answer: 120 3 : (3  2  1)  (3  2  1)  1  24 4 : (4  2  1)  (4  2  1)  3  3  40 9 : (9  2  1)  (9  2  1)  13 13  120 6.

Now it gets dark so mum asked Amy to light up. The lamp is on now but Amy wants to play trick on mum and switch on the lamp 39 times. Is the lamp on/ off now? (Please answer on / off as your final answer) 天已經黑了，現時燈已經開啟了，艾美想捉弄一下媽媽，連按了 39 次燈， 現時燈是亮，還是不亮？（請以亮或不亮作最後答案）

Answer: On / 亮 Switch on the lamp an odd number of times would make the lamp on. 39 is an odd number. 按奇數次會使燈亮，而 39 是個奇數。 奧冠教育出版社 Olympiad Champion Education Publisher 版權所有 不得翻印 All Rights Reserved

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Arithmetic 算術 7.

Find the value of 222  74  788  316 . 求 222  74  788  316 的值。

Answer: 1400

222  74  788  316  (222  788)  (74  316)  1010  390  1400

8.

Find the value of 271113331134 . 求 271113331134 的值。

Answer: 1100

Final Round

34 11  33  13  11 27  34  11  11 3  13  11 27 11 (34  39  27)  11100  1100

9.

Find the value of 11111111 . 求11111111 的值。

Answer: 1,234,321

1 1  1 1111  121 111111  12321 11111111  1234321 10. Find the value of 11  22  33  44  55  66  77  88  99 . 求 11  22  33  44  55  66  77  88  99 的值。 Answer: 495 11  22  33  44  55  66  77  88  99  (11  99)  (22  88)  (33  77)  (44  66)  55  110  110  110  110  55  495 134

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11. Find the value of 416  25  625 . 求 416  25  625 的值。 Answer: 1,000,000 416 25 625  (4 25)  (16 625)  100 10000  1000000

12. Find the value of 2  4  6  8  10 12  14 16  18  20  22  24  26 . 求 2  4  6  8  10 12  14 16  18  20  22  24  26 的值。 Answer: 14

2  4  6  8  10  12  14  16  18  20  22  24  26  2  (6  4)  (10  8)  (14 12)  (18 16)  (22  20)  (26  24)  2222222  14

Number Theory 數論 13. Define the operation symbol a  b  a  b  b  6 , find the value of (17 18) . 定義運算 a  b  a  b  b  6 ，求 (17 18) 的值。 Answer: 318 (17 18)  17 18  18  6  306  18  6  318

14. The difference of A and B is 201. The sum between A and B is 823. Given B is larger than A, find the value of A. A、B 兩數之差是 201，A、B 兩數之和是 823。已知 B 比 A 大，求 A 的值。 Answer: 311 A  (823  201)  2  311 奧冠教育出版社 Olympiad Champion Education Publisher 版權所有 不得翻印 All Rights Reserved

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15. The numbers below follow the arithmetic sequence, what is the sum of the 8th term and the 23rd term? 根據以下的等差數列，問第 8 項及第 23 項之和是甚麼？ 18, 26, 34, 42, 50, … Answer: 268 26  18  8

18  (8  1)  8  18  (23  1)  8  18  56  18  176  268 16. The sum of A and B is 333. B is 8 times of A. Find the value of B. A、B 兩數之和是 333，B 是 A 的 8 倍。求 B 的值。

Final Round

Answer: 296 333  (8  1)  8  37  8  296 17. What is the largest 3-digit number that can be divisible by 3 and 7? 求最大的三位數能同時被 3 及 7 整除。 Answer: 987

3 7  21 1000  21  47...13 1000  13  987 18. The process 「  」has the following results： 已知程序計算「  」有以下結果： 5 1  5 4  2  4  3  12 73  7 65  210 Find the value of (10  4) . 求 (10  4) 的值。 Answer: 5040 (10  4)  10  9  8  7  5040 136

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Geometry 幾何 19. How many square(s) is / are there in the figure below? 請問下圖有多少個正方形？

Question 19 第 19 題 Answer: 14 9  4 1  14 20. A prism has 2018 vertices, how many edge(s) does this prism have? 有一個柱體有 2018 個頂點，問這個柱體有多少條稜？ Answer: 3027 It is a prism with a 1009-sided base. So the prism has 1009  3  3027 edges. 這是個 1009 角柱體，故柱體有 1009  3  3027 條稜。 21. At most how many acute angle(s) could a pentagon (5-sided polygon) contain? 一個五邊形最多可以有多少個銳角？ Answer: 3 89  89  89  273  540 89  89  89  89  184  540 which is not possible 89  89  89  89  184  540 是不可能的。

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22. At least how many squares can be seen if observing the figure below from the top? 如果從上方觀察下圖的立體，最少可以看見多少個正方形？

Question 22 第 22 題 Answer: 22 3 4 3 4 35  22

Final Round

23. At most how many piece(s) can be formed by cutting 5 times on a cake? 在蛋糕上切 5 下，最多可分成多少份？ Answer: 16 1 2  3  4  5 1  16

24. There are two squares in the figure below. Find the perimeter of the figure. 附圖中有 2 個正方形，求附圖的周界。

18

12

Question 24 第 24 題

Answer: 96 18  4  12  12  72  12  12  96

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Combinatorics 組合數學 25. After Amy gives 17 apples to Andy and get 8 apples from Johnny, they have equal number of apples. How many apples more did Johnny have than Andy originally? 當艾美把 17 個蘋果送給安迪及取了強尼 8 個蘋果後，他們擁有的蘋果數目 一樣。問原本強尼比安迪多多少個蘋果？ Answer: 25 17  8  25 26. Choose 3 digits, without repetition, from 1, 2, 4, 6 to construct 3-digit numbers. Of these 3-digit numbers, how many of them are odd numbers? 從 1、2、4、6 中選 3 個不可重複的數位組成三位數。這些三位數當中有 多少個是奇數？ Answer: 6 241, 421, 261,621, 461,641 27. Numbers are drawn from the 35 integers 1 to 35. At least how many numbers are drawn at random to ensure that there are two numbers whose sum is 20? 在 1 至 35 這 35 個整數中最少任意選出多少個，才必定有兩個數之和是 20？ Answer: 27 The worst case is drawing numbers from 20 to 35 and from 1 to 10. Then drawing any one number would suffice. 最壞情況為選出數字 1 至 10 和 20 至 35，之後任意選一個數即可。 28. What is the smallest 4-digit number which is multiple of 5 and 7 and no digits are repeated? 求 5 和 7 的倍數且沒有數位重複最小 4 位數的值。 Answer: 1085 5 7  35

1000  35  28...20 1000  15  1015 1015  35  1050 1050  35  1085 奧冠教育出版社 Olympiad Champion Education Publisher 版權所有 不得翻印 All Rights Reserved

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29. If Andy 29. goes If Andy fromgoes point from B topoint pointB A, to each pointstep A, each can only step can move only up move or move up or right. move right. How many How ways many areways there?are there? 如果安迪從 如果安迪從 B 點走到BA點走到 點，每步只能向上或向右走，共有多少種走法？ A 點，每步只能向上或向右走，共有多少種走法？ A

B

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B

Final Round

Question 29 第 29 題 10 Answer: Answer: 10 1

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30. A teacher 30. Aisteacher distributing is distributing some candies someamong candiesstudents. among students. If the candies If theare candies are distributed distributed evenly among evenly5 among people,52people, candies2remain; candiesifremain; the candies if theare candies are distributed distributed evenly among evenly7 among people,74people, candies4remain; candiesifremain; the candies if theare candies are distributed distributed evenly among evenly9 among people,94people, candies4remain. candiesHow remain. many How candies manydoes candies does the teacher thehave teacher at least? have at least? 老師有若干顆糖果，如果平分給 老師有若干顆糖果，如果平分給 5 人，餘52人，餘 顆；如果平分給 2 顆；如果平分給 7 人，餘74人，餘 4 顆；如果平分給 顆；如果平分給 9 人，餘94人，餘 顆。問老師最少有多少顆糖果？ 4 顆。問老師最少有多少顆糖果？ Answer: Answer: 67 67 4  79  4 67  79  67

67  5  13...2 67  5  13...2

~ End of Paper ~ ~ 全卷完 ~ 140

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Thailand International Thailand International Mathematical Olympiad Mathematical Olympiad FinalMock Round 2017 - 2018 Exam 2017 - 2018 Primary33 Answer KeyKey Primary Answer

Logical Thinking 邏輯思維 5 5 5 5 5 5

Number Theory 數論

1)

5 5

16 2)

5 5

27 3)

5 5

18 4)

5 5

620 5)

5 5

120 6)

5 5

On / 亮

13) 318 14) 311 15) 268 16) 296 17) 987 18) 5040

Arithmetic / Algebra 算術 / 代數 5 5 5 5 5 5

7) 8)

5 5

1100 9)

5 5

1,234,321 10)

5 5

495 11)

5 5

1,000,000 12)

5 5

14

5 5 5 5 5 5 5 5 5 5 5 5

25) 25 26) 6 27) 27 28) 1085 29) 10 30) 67

5 5 5 5 5 5

Geometry 幾何 5 5

1400

Combinatorics 組合數學

19) 14 20) 3027 21) 3 22) 22 23) 16 24) 96

5 5 5 5 5 5

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239

244

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