香 港 國 際 數 學 競 賽 總 決 賽 2 0 2 2

TIONAL MAT
NA H

EM
KON TER

ATICAL OLY
G IN

G MP
HON IAD

Hong Kong International

Mathematical Olympiad
Past Paper Booklet 2022

Final Round

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香港國際數學競賽總決賽 2022
e
HONG KONG INTERNATIONAL
個 MATHEMATICAL OLYMPIAD FINAL ROUND 2022

Primary 4 Question Paper

小學四年級 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 64 chickens and rabbits in a farm. The animals

have a total of 226 legs. How many rabbit(s) is / are there?
農場裡有雞和兔共 64 隻，共 226 條腿。那麼兔有多少隻？

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2. According to the pattern shown below, what number does “?” 5

represent?
按以下規律，求「？」所代表的數字。

12 24 ?
Final Round

Question 2
第2題

3. Bob has 45 marks and Anna has 66 marks in English exam. If the
average mark of Bob, Anna and Mary in the English exam is 57, what
is Mary’s exam mark?
在英文考試中，鮑伯得到了 45 分，安娜得到了 66 分。若鮑伯、
安娜和瑪麗在英文考試的平均分是 57 分，那麼瑪麗在英文考試
中得多少分？

4. 25th July is Tuesday. Which day of the week is 25th September in the
same year?
7 月 25 日是星期二，同一年的 9 月 25 日是星期幾？

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5. A book store had a box of books. One half but 39 fewer books were
sold on the first day. One third of the remaining part and 24 more
books were sold on the second day. Finally, 60 books were left. How
many book(s) in the box was / were originally?
某書店有一箱書本，第一天賣掉了一半少 39 本，第二天賣掉了
剩下的三分之一多 24 本，最後剩下 60 本。問原本一箱書有多少
本書？

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 、 1 、 2 、 3 、 4 、 6 、 9 、 13 、 19 、 __ 、….

at

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Arithmetic
算術

7. Find the value of 520 13 + 520  65 + 520  26 . 1

求 520 13 + 520  65 + 520  26 的值。

8. What should be the number filled in the blank if the equation below is
correct? 1
在以下算式填上甚麼數字會使算式正確？
( ___ + 8)  3 + 5 =209
1
9. Find the value of 3 + 9 + 15 + 21 + 27 + + 141 + 147 + 153 +159 .
求 3 + 9 + 15 + 21 + 27 + + 141 + 147 + 153 +159 的值。
Final Round

1
10. Find the value of .
2 1
1+
3
4+
5
1
求 的值。
2
1+ 1
3
4+
5

11. Find the value of 101011010100 .

求 101011010100 的值。 1

12. Find the value of 583 11 + 720  99 − 363 11 .

求 583 11 + 720  99 − 363 11 的值。

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Number Theory
數論

13. The sum of positive integers A and B is 672. A is 5 times of B. Find the
value of A.
兩個正整數 A、B 之和是 672，A 是 B 的 5 倍。求 A 的值。
is
14. How many factors does 14700 have?
14700 有多少個因數？

15. Define the operation symbol a  b = ( a  b )  ( a + b ) , find the value of

6  ( 4  7) .
定義運算符號 a  b = ( a  b )  ( a + b ) ，求 6  ( 4  7 ) 的值。

16. Find the unit digit of A if A = 3  3  3  ...  3  7  7  7  ... 7 .

20' s 19' s
若 A = 3  3  3  ...  3  7  7  7  ... 7 ，求 A 的個位數的值。
20' s 19' s

17. How many 3-digit number(s) either divisible by 4 or divisible by 9 is /

are there?
有多少個三位數能被 4 或 9 整除？

18. If a 10-digit number 202206105A is divisible by 9, find the largest

possible 1-digit number represented by A.
若十位數 202206105A 可被 9 整除，求 A 所代表的最大可能一位
數字。

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Geometry 2
幾何

19. How many rectangle(s) containing “*” is / are there in the figure
below?
請問下圖有多少個含有「*」的矩形？

*
Final Round

Question 19
第 19 題

20. The perimeter of a rectangle is 396. If lengths of sides of the rectangle

are integers, find the maximum possible value of the area of the
rectangle.
一個矩形的周界為 396，其邊長是整數，求矩形面積的最大可能
值。

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21. How many rectangle(s) is / are there in the figure below?

請問下圖有多少個長方形？

Question 21
第 21 題

le

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22. At least how many square(s) can be seen if viewing the figure below 2
from the right?
如果從側右面觀察下圖的立體，最少可以看見多少個正方形？
Final Round

Question 22
第 22 題
2
23. A prism has 72 vertices. Find the sum of the number of edges and the
number of faces of this prism.
有一個柱體有 72 個頂點，求這個柱體棱和面的數量之和。

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24. How many right-angled triangle(s) is / are there in the figure below?
下圖中有多少個直角三角形？

Question 24
第 24 題

Combinatorics
組合數學

25. A teacher prepared 79 masks to give to 8 students. At least how many

e mask(s) does the student who gets the most masks have?
老師準備了 79 個口罩派發給 8 名學生，擁有最多口罩的學生最
少得到多少個口罩？

26. How many 2-digit number(s) is / are there such that the number does
not contain digit “2”?
有多少個不包含數位 2 的兩位數？

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27. A flight of stairs has 9 steps. Anthony can climb up 1 step or 2 steps
at a time. How many way(s) is / are there for Anthony to go up the
stairs?
一道有 9 級的樓梯，安東尼每一步可以上 1 級或 2 級，問安東尼
走上這道樓梯共有多少種方法？

28. Adam has 30 $1, $2 and $5 coins altogether. He has $123 in total. At
most how many $1 coin(s) does he have?
亞當有 1 元、2 元和 5 元硬幣共 30 個，合共 123 元。問亞當最多
有多少個 1 元硬幣？ 填
O
29. Numbers are drawn from 80 integers among 19 to 98. At least how p
many number(s) needed to be drawn to ensure that there are two
numbers drawn whose product is divisible by 22?
Final Round

在 19 至 98 這 80 個整數中最少需要選出多少個數，才必定有兩
個數之積能被 22 整除？
1
30. How many 3-digit number(s) is / are there such that each of the
number(s) is greater than 500 which does not contain digit “0”, “3” or
“9”?
有多少個不包含數位 0、3 或 9 且大於 500 的三位數？
2

~ 全卷完 ~
~ End of Paper ~

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香港國際數學競賽總決賽 2022
HONG KONG INTERNATIONAL
MATHEMATICAL OLYMPIAD FINAL ROUND 2022

Primary 4 Detailed Solution

小學四年級 詳細題解

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
個 point for wrong answer)

Logical Thinking
邏輯思維

1. There are a total of 64 chickens and rabbits in a farm. The animals

have a total of 226 legs. How many rabbit(s) is / are there?
農場裡有雞和兔共 64 隻，共 226 條腿。那麼兔有多少隻？
e
解：49
個 ( 226 − 2  64 )  ( 4 − 2 )
= ( 226 − 128 )  2
= 98  2
= 49

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2. According to the pattern shown below, what number does “?” 3

represent?
按以下規律，求「？」所代表的數字。

4
12 24 ?
Final Round

Question 2
第2題 解

解：20
LCM ( 6, 4 ) = 12
LCM ( 8,12 ) = 24
5
LCM ( 4,10 ) = 20
 ? =20

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3. Bob has 45 marks and Anna has 66 marks in English exam. If the
average mark of Bob, Anna and Mary in the English exam is 57, what
is Mary’s exam mark?
在英文考試中，鮑伯得到了 45 分，安娜得到了 66 分。若鮑伯、
安娜和瑪麗在英文考試的平均分是 57 分，那麼瑪麗在英文考試
中得多少分？

解：60
57  3 − 45 − 66 = 126 − 66 = 60

4. 25th July is Tuesday. Which day of the week is 25th September in the
same year?
7 月 25 日是星期二，同一年的 9 月 25 日是星期幾？

解： Mon / 一
2 + ( 31 − 25 ) + 31 + 25 =
64
64  7 = 9 1
 Mon / 一 [作者姓氏] / "[從標題開始的 1 到 2 個字]" / 205

5. A book store had a box of books. One half but 39 fewer books were
sold on the first day. One third of the remaining part and 24 more
books were sold on the second day. Finally, 60 books were left. How
many book(s) in the box was / were originally?
某書店有一箱書本，第一天賣掉了一半少 39 本，第二天賣掉了
剩下的三分之一多 24 本，最後剩下 60 本。問原本一箱書有多少
本書？

解：174
 3 
( 60 + 24 )  2 − 39   2
= (126 − 39 )  2
= 174

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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 、 1 、 2 、 3 、 4 、 6 、 9 、 13 、 19 、 __ 、…. 7

解：28
1+ 0 = 1:1 解
1+ 0 = 1: 2
1+ 0 = 1: 3
1+1= 2:4
1+ 2 = 3:5 8
1+ 3 = 4:6
Final Round

2+4= 6:7
3+6 = 9:8
4+9= 13 : 9
解
6 + 13 = 19 :10
 9 + 19 = 28 :11

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Arithmetic
算術

7. Find the value of 520 13 + 520  65 + 520  26 .

求 520 13 + 520  65 + 520  26 的值。

解：68
520 13 + 520  65 + 520  26
= 40 + 8 + 20
= 68

8. What should be the number filled in the blank if the equation below is
correct?
在以下算式填上甚麼數字會使算式正確？
( ___ + 8)  3 + 5 =209
解：60
( 209 − 5 )  3 − 8
= ( 204 )  3 − 8
= 68 − 8
= 60

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9. Find the value of 3 + 9 + 15 + 21 + 27 + + 141 + 147 + 153 +159 . 1

求 3 + 9 + 15 + 21 + 27 + + 141 + 147 + 153 +159 的值。

解：2187 解
3 + 9 + 15 + 21 + 27 + + 141 + 147 + 153 + 159
= (159 + 3)  (159 − 3)  6 + 1  2
= (162  27 )  2
= 2187

1
10. Find the value of .
2
1+
3 1
4+
Final Round

5
1
求 的值。
2
1+ 解
3
4+
5

23
解：
33
11 1 1 23
= = = =
2 2 10 33 33
1+ 1+ 1+
3 23 23 23
4+
5 5

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11. Find the value of 101011010100 .

求 101011010100 的值。

解：10203020100
101101 =10201
1010110101 = 102030201
101011010100
= 1010110101100
= 102030201100
= 10203020100

12. Find the value of 583 11 + 720  99 − 363 11 .

求 583 11 + 720  99 − 363 11 的值。

3
解： 27
11
583  11 + 720  99 − 363  11
80
= 53 + − 33
11
3
=53 + 7 − 33
11
3
= 27
11

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Number Theory 1
數論

13. The sum of positive integers A and B is 672. A is 5 times of B. Find the
value of A.
兩個正整數 A、B 之和是 672，A 是 B 的 5 倍。求 A 的值。 解

解：560
672  ( 5 + 1)  5
= 112  5
= 560

14. How many factors does 14700 have?

Final Round

14700 有多少個因數？
1
解：108
14700 = 22  3  52  7 2
 2  ( 2 + 1)  (1 + 1)  ( 2 + 1)  ( 2 + 1) = 2  3  2  3  3 =108

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15. Define the operation symbol a  b = ( a  b )  ( a + b ) , find the value of

6  ( 4  7) .
he 定義運算符號 a  b = ( a  b )  ( a + b ) ，求 6  ( 4  7 ) 的值。

解：580272
6  ( 4  7 ) =6  ( ( 4  7 )  ( 4 + 7 ) )
6 ( 28 11)
=
= 6  308
= ( 6  308 )  ( 6 + 308 )
= 1848  314
= 580272

16. Find the unit digit of A if A = 3  3  3  ...  3  7  7  7  ... 7 .

20' s 19' s
若 A = 3  3  3  ...  3  7  7  7  ... 7 ，求 A 的個位數的值。
20' s 19' s

解：3
A = 3  3  3  ...  3  7  7  7  ...  7
20' s 19' s

A = 21 21 21 ...  21 3

19' s

1 3 =3

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17. How many 3-digit number(s) either divisible by 4 or divisible by 9 is /

are there?
有多少個三位數能被 4 或 9 整除？
1
解：300
999  4 = 249...3
99  4 = 24...3
999  9 = 111
99  9 =11
999  36 = 27...27
99  36 = 2...27
 (249 − 24) + (111 − 11) − (27 − 2)
Final Round

= 225 + 100 − 25
= 300

18. If a 10-digit number 202206105A is divisible by 9, find the largest

possible 1-digit number represented by A.
若十位數 202206105A 可被 9 整除，求 A 所代表的最大可能一位 解
數字。

解：9
2 + 0 + 2 + 2 + 0 + 6 + 1 + 0 + 5 + A = 18 + A = 18 / 27
A=0/9
A=
9

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s/ Geometry
幾何

19. How many rectangle(s) containing “*” is / are there in the figure
below?
請問下圖有多少個含有「*」的矩形？

Question 19
第 19 題

解：80
5 2  2  4 =80

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20. The perimeter of a rectangle is 396. If lengths of sides of the rectangle 2

are integers, find the maximum possible value of the area of the
rectangle.
一個矩形的周界為 396，其邊長是整數，求矩形面積的最大可能
值。

解：9801
396  4 = 99
 99  99 =9801 [作者姓氏] / "[從標題開始的 1 到 2 個字]" / 215

21. How many rectangle(s) is / are there in the figure below?

請問下圖有多少個長方形？
Final Round

Question 21
第 21 題

解：126
(1 + 2 + 3)  (1 + 2 + 3 + 4 + 5 + 6 )
= 6  21
= 126

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le 22. At least how many square(s) can be seen if viewing the figure below
from the right?
如果從側右面觀察下圖的立體，最少可以看見多少個正方形？
能

15

Question 22
第 22 題

解：8

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23. A prism has 72 vertices. Find the sum of the number of edges and the
number of faces of this prism.
有一個柱體有 72 個頂點，求這個柱體棱和面的數量之和。
2
解：146
E : ( 72  2 )  3 =108
F : ( 72  2 ) + 2 =38
108 + 38 =
146 [作者姓氏] / "[從標題開始的 1 到 2 個字]" / 218 解

24. How many right-angled triangle(s) is / are there in the figure below?
下圖中有多少個直角三角形？
2
Final Round

Question 24
第 24 題

解：8 解
4+4 =8

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e Combinatorics
組合數學

25. A teacher prepared 79 masks to give to 8 students. At least how many

mask(s) does the student who gets the most masks have?
老師準備了 79 個口罩派發給 8 名學生，擁有最多口罩的學生最
少得到多少個口罩？

18 解：10
79  8 =9...7
10

26. How many 2-digit number(s) is / are there such that the number does
not contain digit “2”?
有多少個不包含數位 2 的兩位數？

解：72
8 9 =72

27. A flight of stairs has 9 steps. Anthony can climb up 1 step or 2 steps
at a time. How many way(s) is / are there for Anthony to go up the
stairs?
一道有 9 級的樓梯，安東尼每一步可以上 1 級或 2 級，問安東尼
走上這道樓梯共有多少種方法？

解：55
1st 2nd 3rd 4th 5th 6th 7th 8th 9th

1 2 3 5 8 13 21 34 55

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 Hong Kong International Mathematical Olympiad Final Round 2022
[作者姓氏] / "[從標題開始的 1 到 2 個字]" / 220

28. Adam has 30 $1, $2 and $5 coins altogether. He has $123 in total. At
most how many $1 coin(s) does he have?
亞當有 1 元、2 元和 5 元硬幣共 30 個，合共 123 元。問亞當最多
有多少個 1 元硬幣？

解：6
30  5 =150
150 − 123 = 27
27 = 4 + 4 + 4 + 4 + 4 + 4 + 3 填
6 O
p
29. Numbers are drawn from 80 integers among 19 to 98. At least how
many number(s) needed to be drawn to ensure that there are two
Final Round

numbers drawn whose product is divisible by 22?

在 19 至 98 這 80 個整數中最少需要選出多少個數，才必定有兩
個數之積能被 22 整除？ 1

解：74
11: 22,33, 44,..., 77,88 : 7
80 − 7 + 1 =74
解
30. How many 3-digit number(s) is / are there such that each of the
number(s) is greater than 500 which does not contain digit “0”, “3” or
“9”?
有多少個不包含數位 0、3 或 9 且大於 500 的三位數？

解：196
4  7  7 = 28  7 = 196
~ 全卷完 ~
~ End of Paper ~

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Hong Kong International

Mathematical Olympiad
Final Round 2022
Primary 4 Answer Key
Logical Thinking Number Theory Combinatorics
邏輯思維 數論 組合數學
1) 13) 25)
5 5 5 5 5 5 5
49 560 10
2) 14) 26)
5 5 5 5 5 5
5 20 108 72
3) 15) 27)
5 5 5 5 5 5
5 60 580,272 55
4) 16) 28)
5 5 5 5 5 5
5 Mon / 一 3 6
5) 17) 29)
5 5 5 5 5 5
5 174 300 74
6) 18) 30)
5 5 5 5 5 5
5 28 9 196
Arithmetic / Algebra Geometry
算術 / 代數 幾何
7) 19)
5 5 5 5
68 80
8) 20)
5 5 5 5
60 9801
9) 21)
5 5 5 5
2187 126
23 22)
5 10) 5 5 8 5
33
11) 23)
5 5 5 5
10,203,020,100 146
3 24)
5 12) 27 5 5 8 5
11

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