Timo Final 20-21 P4 [PDF]

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Thailand International Mathematical Olympiad Past Paper Booklet 2020 – 2021 Final Round 奧冠教育出版社 Olympiad Champion Education Publisher 版權所有 不得翻印 All Rights Reserved

1

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 66

65

泰國國際數學競賽總決賽 2020 - 2021 THAILAND INTERNATIONAL MATHEMATICAL OLYMPIAD FINAL ROUND 2020 - 2021

h e

。

Primary 4 小學四年級

Question Paper 試題

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.

There are a total of 41 chickens and rabbits in a farm. The animals have a total of 96 legs. How many chicken(s) is / are there? 農場裡有雞和兔共 41 隻，共有 96 條腿。那麼雞有多少隻？

2.

Edward is now playing “Clapping Game”. When he needs to call any multiples of 6, he has to clap hands once instead of calling them out. Now the game starts from 20 in an ascending order. After clapping 37 times, what will the next number be? 愛德華現在玩「拍手」遊戲，當他需要讀出任何 6 的倍數時，他 必須以拍手代替直接讀出該數字。現時遊戲由 20 開始以順數形 式，當拍手 37 次後，求下一個數的值。

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65

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 67 3.

In class 4B, all students queue up to form a rectangle. On Eric’s right and left hand side, there are 8 and 5 students respectively. There are 6 students in front of and 5 students behind Eric respectively. How many student(s) is / are in class 4B? 在 4B 班，所有學生排隊排成一個矩形，在艾力的右手方 8 名學 生，左手方有 5 名學生，前面有 6 名學生，後面有 5 名學生。問 4B 班有多少名學生？

Final Round

4.

25th March 2021 is Thursday. Which day of the week is 24th November that year? 2021 年 3 月 25 日是星期四，同年 11 月 24 日是星期幾？

5.

According to the pattern shown below, find the value of B − A . 按以下規律，求 B − A 的值。 3

4

5

6

7

8

9

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15

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21

24

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30

45

54

63

A

81

162

189

B

10

7

8

11

567 6.

66

Peter’s uncle’s age this year minuses 29, then divided by 6, adds 16 and multiplies by 8. The result will be 152 years old. How old is Peter’s uncle this year? 彼得的叔叔今年的年齡，把它減去 29 之後除以 6，再加上 16 後 再變成 8 倍，結果是 152 歲，那麼彼得的叔叔今年多少歲？

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9

1

67

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 68

t 6

學 問

後

Arithmetic 算術 7. Find the value of

1

求 2+

8.

8

9+

1 2+

8

9+

. 4 7

的值。 4 7

If A and B are both 1-digit numbers, what is the value of A + B if the equation with carrying is correct? 若 A 和 B 均為一位數字，且以下有進位的算式正確，求 A + B 的 值。

 5

A

A

1

B

A

7

0

9

Question 8 第8題 9.

Find the value of 9 1113 15 . 求 9 1113 15 的值。

10. Find the value of 208 + 222 + 236 + 250 + ... + 530 + 544 . 求 208 + 222 + 236 + 250 + ... + 530 + 544 的值。

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67

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 69 [作者姓氏] / "[從標題開始的 1 到 2 個字]" / 69 11. Find the value of 592  23 −12  23 + 97  23 + 174  23 . 求 592  23 −12  23 + 97  23 + 174  23 的值。 11. Find the value of 592  23 −12  23 + 97  23 + 174  23 . 求 592  23 −12  23 + 97  23 + 174  23 的值。 12. Using method = S 2S − S , find the value of S = 7 + 14 + 28 + ... + 1792 + 3584 . 12. Using method = S 2S − S , find the value of 使用方法 = S 2S − S ，求 S = 7 + 14 + 28 + ... + 1792 + 3584 的值。 S = 7 + 14 + 28 + ... + 1792 + 3584 . 使用方法 = S 2S − S ，求 S = 7 + 14 + 28 + ... + 1792 + 3584 的值。

Final Round

Number Theory 數論 Number Theory 13. If a 10-digit number 202 A89293 B is divisible by 28 and B  A  0 , 數論 find the value of A. 13. If a 10-digit number 202 A89293B is divisible by 28 and B  A  0 , 若十位數 202 A find the value of89293 A. B 可被 28 整除且 B  A  0 ，求 A 的值。 若十位數 202 A89293B 可被 28 整除且 B  A  0 ，求 A 的值。 14. The sum of 9 consecutive even numbers is 1494. Find the largest number. 14. The sum of 9 consecutive even numbers is 1494. Find the largest 9 個連續偶（雙）數之和為 1494，求最大的數。 number. 9 個連續偶（雙）數之和為 1494，求最大的數。 15. The product of positive numbers A and B is 1014. A is 6 times of B. Find the value of A. 15. The product of positive numbers A and B is 1014. A is 6 times of B. 兩個正整數 A、B 的積是 1014，A 是 B 的 6 倍。求 A 的值。 Find the value of A. 兩個正整數 A、B 的積是 1014，A 是 B 的 6 倍。求 A 的值。 16. Define the operation symbol a  b = (b − a)  (a + b) − (a + a − b) , find 29 . a  b = (b − a)  (a + b) − (a + a − b) , find value ( 6  8 ) symbol 16. the Define theofoperation 定義運算 a the value of ( 6b=8(b) − a29) . (a + b) − (a + a − b) ，求 ( 6  8 )  29 的 值。 定義運算 a  b = (b − a)  (a + b) − (a + a − b) ，求 ( 6  8 )  29 的 值。

68

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1

1

1

69

69

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 70 17. Find the unit digit of A if

A = 3  3  3  ...  3  4  4  4  ...  4  7  7  7  ...  7 . 2021' s

2021' s

2021' s

若 A = 3  3  3  ...  3  4  4  4  ...  4  7  7  7  ...  7 ，求 A 的個位 2021' s

2021' s

2021' s

數的值。 18. What is the largest 3-digit number that can be divisible by 21 and 45? 求最大的三位數能同時被 21 及 45 整除。

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

nd

nd Question 19 第 19 題

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69

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 71 20. It is given that the area of a rectangle is 336. The width and length are integers. Find the minimum value of the perimeter. 已知一個矩形面積為 336，而長和闊均是整數，求周界的最小 值。 21. The figure on the left is a small right-angled triangle with side length 8 and hypotenuse 10. The larger triangle on the right is formed by 4 identical small right-angled triangles. Find the area of the large triangle on the right. 左圖為一個邊長為 8 及斜邊為 10 的小直角三角形。右圖中的大 三角形由 4 個相同的小直角三角形組成。求大三角形的面積。

Final Round

10 8 圖1 Figure 1

Question 21 第 21 題

70

圖2 Figure 2

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2

71

re

h

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 72 22. There are 3 levels in the figure below and it consists of 9 cubes. According to the following pattern, how many cube(s) will be needed to build 12 levels? 下圖中的立體有 3 層，由 9 個正方體組成。根據下圖規律，請問 需要用多少個正方體來堆砌成 12 層？

Question 22 第 22 題

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71

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 73 23. How many rectangle(s) with “*” is / are there in the figure below? 請問下圖有多少個含有「*」的矩形？

2

*

Final Round

Question 23 第 23 題 24. The perimeter of a square is 20. Now Peter combines 25 squares to a new rectangle. The side lengths of the new rectangle are integers. What is the maximum value of the perimeter of the rectangle? 1 個正方形的周界為 20，現時彼德把 25 個正方形組合成 1 個新 的矩形，新矩形的邊長為整數。求新矩形的周界的最大值。

2

2

2

2

72

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73

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 74 Combinatorics 組合數學 25. Now there are infinitely many boxes. If 2800 candies are needed to separate into these boxes evenly, how many way(s) is / are there? 現有無限多個箱子，若要把 2800 粒糖果平均分配到進箱子，問有 多少種方法？ 26. Numbers are drawn from 85 integers 46 to 130. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose difference is 17? 在 46 至 130 這 85 個整數中最少任意選出多少個數，才必定有兩 個數之差是 17？ 27. A flight of stairs has 11 steps. Andy can go up for 1 step or 2 steps each time. The 6th step cannot be stepped on as it is destroyed. How many way(s) is / are there for Andy to go up the stairs? 一道有 11 級的樓梯，安迪每一步可以上 1 級或 2 級，其中第 6 級因為損壞而不能踏上，問安迪走上這道樓梯共有多少種方法？ 28. Find the smallest difference by using 2, 2, 3, 4, 4, 7, 7, 7, 9, 9 to form two 5-digit numbers. 求使用 2、2、3、4、4、7、7、7、9、9 來組成 2 個五位數之差的 最小值。 29. 83 biscuits are either in box A, B, C, D or E. It would be at least one biscuit in the box. In the box containing most biscuits, at least how many biscuit(s) is / are in the box? 83 塊餅乾分到盒子 A、B、C、D 或 E 中，若每盒子至少有 1 塊 餅乾。在最多餅乾的那個盒子，至少會有多少塊餅乾？

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73

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 75 30. It is known that the most-right digit smaller than any other digit(s) in a number is called “special number”. For example, 831 and 554. How many 3-digit “special number” is / are there? 已知最右數位比任何其他數位都小的數稱為「特別數」，例如： 831 和 554，問有多少個三位的「特別數」？ ~ 全卷完 ~ ~ End of Paper ~

填 O po



Final Round

1.

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[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 200

泰國國際數學競賽總決賽 2020 - 2021 THAILAND INTERNATIONAL MATHEMATICAL OLYMPIAD FINAL ROUND 2020 - 2021 Primary 4 小學四年級

Detailed Solution 詳細解答

Final Round

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)

2

解

3

Logical Thinking 邏輯思維 1.

There are a total of 41 chickens and rabbits in a farm. The animals have a total of 96 legs. How many chicken(s) is / are there? 農場裡有雞和兔共 41 隻，共有 96 條腿。那麼雞有多少隻？

解 解：34

7 ( 96 − 41 2 )  (4 − 2) = 41 − 7 = 34

194

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00

y

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 201 2.

Edward is now playing “Clapping Game”. When he needs to call any multiples of 6, he has to clap hands once instead of calling them out. Now the game starts from 20 in an ascending order. After clapping 37 times, what will the next number be? 愛德華現在玩「拍手」遊戲，當他需要讀出任何 6 的倍數時，他 必須以拍手代替直接讀出該數字。現時遊戲由 20 開始以順數形 式，當拍手 37 次後，求下一個數的值。

解：241

24,30, , 234, 240  241 3.

In class 4B, all students queue up to form a rectangle. On Eric’s right and left hand side, there are 8 and 5 students respectively. There are 6 students in front of and 5 students behind Eric respectively. How many student(s) is / are in class 4B? 在 4B 班，所有學生排隊排成一個矩形，在艾力的右手方 8 名學 生，左手方有 5 名學生，前面有 6 名學生，後面有 5 名學生。問 4B 班有多少名學生？

解：168

(8 + 1 + 5)  ( 6 + 1 + 5) =

14  12 = 168

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195

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 202 4.

25th March 2021 is Thursday. Which day of the week is 24th November that year? 2021 年 3 月 25 日是星期四，同年 11 月 24 日是星期幾？

6

解：Wed / 三

(31 − 25) + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 24 = 244 244  7 = 34...6

解

 Wed / 三 5.

According to the pattern shown below, find the value of B − A . 按以下規律，求 B − A 的值。

Final Round

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27

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63

A

81

162

189

B

10

11

7

567 解：144 A = 21 + 24 + 27 = 72 B = 63 + 72 + 81 = 216  B − A= 216 − 72= 144

196

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解

02

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 203 6.

Peter’s uncle’s age this year minuses 29, then divided by 6, adds 16 and multiplies by 8. The result will be 152 years old. How old is Peter’s uncle this year? 彼得的叔叔今年的年齡，把它減去 29 之後除以 6，再加上 16 後 再變成 8 倍，結果是 152 歲，那麼彼得的叔叔今年多少歲？

解：47

(152  8 −16)  6 + 29 = 3  6 + 29 = 47 Arithmetic 算術 7. Find the value of

1

求 2+

解：

8

9+

1 2+

8

9+

. 4 7

的值。 4 7

67 190 1 1 1 1 67 = = = = 8 8 8  7 67  2 + 56 190 2+ 2+ 2+ 4 97 + 4 67 67 9+ 7 7

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197

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 204 8.

If A and B are both 1-digit numbers, what is the value of A + B if the equation with carrying is correct? 若 A 和 B 均為一位數字，且以下有進位的算式正確，求 A + B 的 值。

 5

A

A

1

B

A

7

0

9

Question 8 第8題

Final Round

解：10 A= 1/ 3 / 9 A = 9, B = 0 : 99  109 = 10791 A = 1, B = 9 :11 191 = 2101 A = 3, B = 0 : 33  103 = 3399 A = 3, B = 9 : 33  193 = 6369 A= 3 5709  33 = 173  A + B = 3 + 7 = 10

9

解

1

解

1

解

198

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[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 205

04

e

9.

Find the value of 9 1113 15 . 求 9 1113 15 的值。

的 解：19305 9  11 13  15 = 99  13  15 = 1287  15 = 19305 10. Find the value of 208 + 222 + 236 + 250 + ... + 530 + 544 . 求 208 + 222 + 236 + 250 + ... + 530 + 544 的值。 解：9400 208 + 222 + 236 + 250 + ... + 530 + 544 (208 + 544)  25 = 2 752  25 = 2 = 9400 11. Find the value of 592  23 −12  23 + 97  23 + 174  23 . 求 592  23 −12  23 + 97  23 + 174  23 的值。 解：37 592  23 − 12  23 + 97  23 + 174  23= (592 − 12 + 97 + 174)  23 = 851  23 = 37

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199

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 206 12. Using method = S 2S − S , find the value of S = 7 + 14 + 28 + ... + 1792 + 3584 . 使用方法 = S 2S − S ，求 S = 7 + 14 + 28 + ... + 1792 + 3584 的值。

1

解：7161 S = 7 + 14 + 28 + ... + 1792 + 3584 2S = 14 + 28 + 56 + ... + 3584 + 7168 S = 2S − S = 7168 − 7 = 7161

解

Number Theory 數論

Final Round

13. If a 10-digit number 202 A89293B is divisible by 28 and B  A  0 , find the value of A. 若十位數 202 A89293B 可被 28 整除且 B  A  0 ，求 A 的值。 解：1 B = 2/6

936 − 892 + (20 + A) − 2 = 62 + A A= 1/ 8 932 − 892 + (20 + A) − 2 = 58 + A A= 5  B= 6, A= 1  2021892936

200

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1

解

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 207

06

14. The sum of 9 consecutive even numbers is 1494. Find the largest number. 9 個連續偶（雙）數之和為 1494，求最大的數。 解：174 1494  9 = 166 166 + 2 + 2 + 2 + 2 = 174

15. The product of positive numbers A and B is 1014. A is 6 times of B. Find the value of A. 兩個正整數 A、B 的積是 1014，A 是 B 的 6 倍。求 A 的值。 解：78 1041  6 = 169 169= 13 13  6 13 =78

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201

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 208 16. Define the operation symbol a  b = (b − a)  (a + b) − (a + a − b) , find the value of ( 6  8 )  29 .

1

定義運算 a  b = (b − a)  (a + b) − (a + a − b) ，求 ( 6  8 )  29 的 值。 解：246 ( 6  8)  29 =

=

( (8 − 6)  (6 + 8) − (6 + 6 − 8) )  29 ( 28 − 4 )  29

解

Final Round

= 24  29 = (29 − 24)  (24 + 29) − (24 + 24 − 29) =5  53 − 19 = 246

1

解

202

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[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 209

08

17. Find the unit digit of A if

A = 3  3  3  ...  3  4  4  4  ...  4  7  7  7  ...  7 . 2021' s

2021' s

2021' s

若 A = 3  3  3  ...  3  4  4  4  ...  4  7  7  7  ...  7 ，求 A 的個位 2021' s

2021' s

2021' s

數的值。 解：4 A = 3  3  3  ...  3  4  4  4  ...  4  7  7  7  ...  7 2021' s

2021' s

2021' s

A = 84  84  84  ...  84 2021'2

4  4= 16,1 4= 4  2021  2 = 1010...1 4 18. What is the largest 3-digit number that can be divisible by 21 and 45? 求最大的三位數能同時被 21 及 45 整除。 解：945 L.C.M ( 21, 45 ) = 5  9  7 = 315

999  315 = 3...54  315  3 = 945

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203

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 210 Geometry 幾何

2

19. How many rectangle(s) is / are there in the figure below? 請問下圖有多少個長方形？

解

2

Final Round

Question 19 第 19 題 解：73 (1 + 2 + 3 + 4 + 5)  (1 + 2) + (1 + 2)  (1 + 2 + 3 + 4) + 1 (1 + 2 + 3 + 4) − (1 + 2)  (1 + 2) − 1 (1 + 2) = 15  3 + 3 10 + 110 − 3  3 − 1 3 = 45 + 30 + 10 − 9 − 3 = 73

204

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[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 211

10

20. It is given that the area of a rectangle is 336. The width and length are integers. Find the minimum value of the perimeter. 已知一個矩形面積為 336，而長和闊均是整數，求周界的最小 值。 解：74 336 = 24  3  7 = 16  21  (16 + 21)  2 = 74

21. The figure on the left is a small right-angled triangle with side length 8 and hypotenuse 10. The larger triangle on the right is formed by 4 identical small right-angled triangles. Find the area of the large triangle on the right. 左圖為一個邊長為 8 及斜邊為 10 的小直角三角形。右圖中的大 三角形由 4 個相同的小直角三角形組成。求大三角形的面積。 10 8 圖1 Figure 1

Question 21 第 21 題

圖2 Figure 2

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205

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 212

2

解：96 2

2

10 − 8 = 6 8 6 96 4 = 2

22. There are 3 levels in the figure below and it consists of 9 cubes. According to the following pattern, how many cube(s) will be needed to build 12 levels? 下圖中的立體有 3 層，由 9 個正方體組成。根據下圖規律，請問 需要用多少個正方體來堆砌成 12 層？

Final Round

解

2

Question 22 第 22 題 解：144 1 + 3 + 5 + 7 + ... + 19 + 21 + 23 = 24 12  2 = 144

206

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解

12

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 213 23. How many rectangle(s) with “*” is / are there in the figure below? 請問下圖有多少個含有「*」的矩形？

d

問

*

Question 23 第 23 題 解：96 4  3 2  4 = 96

24. The perimeter of a square is 20. Now Peter combines 25 squares to a new rectangle. The side lengths of the new rectangle are integers. What is the maximum value of the perimeter of the rectangle? 1 個正方形的周界為 20，現時彼德把 25 個正方形組合成 1 個新 的矩形，新矩形的邊長為整數。求新矩形的周界的最大值。 解：260 20  4 = 5

5  5  25 = 625 = 5 125  ( 5 + 125)  2 = 260

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207

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 214 Combinatorics 組合數學 25. Now there are infinitely many boxes. If 2800 candies are needed to separate into these boxes evenly, how many way(s) is / are there? 現有無限多個箱子，若要把 2800 粒糖果平均分配到進箱子，問有 多少種方法？

2

解

解：30 2800 = 2  2  2  2  5  5  7

 ( 4 + 1)  ( 2 + 1)  (1 + 1) =5  3  2 =30

Final Round

26. Numbers are drawn from 85 integers 46 to 130. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose difference is 17? 在 46 至 130 這 85 個整數中最少任意選出多少個數，才必定有兩 個數之差是 17？ 解：52 {46, 47,..., 62}:17 {63, 64,..., 79}:17 {80,81,...,96}:17 {97,98,...,113}:17 {114,115,...,130}:17 17 + 17 + 17 + 1 = 52

208

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2

解

2

解

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 215

14

有

兩

27. A flight of stairs has 11 steps. Andy can go up for 1 step or 2 steps each time. The 6th step cannot be stepped on as it is destroyed. How many way(s) is / are there for Andy to go up the stairs? 一道有 11 級的樓梯，安迪每一步可以上 1 級或 2 級，其中第 6 級因為損壞而不能踏上，問安迪走上這道樓梯共有多少種方法？ 解：40 1st

2nd

3rd

4th

5th

1

2

3

5

8

6th

7th

8th

9th

10th 11th

8

8

16

24

40

28. Find the smallest difference by using 2, 2, 3, 4, 4, 7, 7, 7, 9, 9 to form two 5-digit numbers. 求使用 2、2、3、4、4、7、7、7、9、9 來組成 2 個五位數之差的 最小值。 解：4 92747 − 92743 = 4

29. 83 biscuits are either in box A, B, C, D or E. It would be at least one biscuit in the box. In the box containing most biscuits, at least how many biscuit(s) is / are in the box? 83 塊餅乾分到盒子 A、B、C、D 或 E 中，若每盒子至少有 1 塊 餅乾。在最多餅乾的那個盒子，至少會有多少塊餅乾？ 解：17 83  5 = 16...3 17

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209

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 216 30. It is known that the most-right digit smaller than any other digit(s) in a number is called “special number”. For example, 831 and 554. How many 3-digit “special number” is / are there? 已知最右數位比任何其他數位都小的數稱為「特別數」，例如： 831 和 554，問有多少個三位的「特別數」？ 解：204 111 + 1 2  2 + 1 3  3 + + 1 6  6 + 1 7  7 + 1 8  8 =1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204

Final Round

~ 全卷完 ~ ~ End of Paper ~



填 O p

1

解

210

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25

[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 326

Thailand International Mathematical Olympiad Final Round 2020 - 2021 Primary 4 Answer Key Logical Thinking 邏輯思維

5

5

5

5

5

5

5

5

5

5

5

5

5 5 5 5 5 5

1) 2) 3) 4) 5) 6)

34 241 168 Wed / 三 144

47 Arithmetic / Algebra 算術 / 代數 7) 8) 9) 10) 11) 12)

67 190 10 19305 9400 37 7161

Number Theory 數論 5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

5

13) 14) 15) 16) 17) 18)

19) 20) 21) 22) 23) 24)

1 174 78 246 4 945 Geometry 幾何 73 74 96 144 96 260

Combinatorics 組合數學 5

5

5

5

5

5

5

5

5

5

5

5

25) 26) 27) 28) 29) 30)

30 52 40 4 17 204

5 5 5 5 5 5

5 5 5 5 5 5

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317