奧 冠 教 育 中 心

OLYMPIAD CHAMPION EDUCATION CENTRE

Room 309-310, 8 Jordan Road, Yau Ma Tei, Kowloon, Hong Kong SAR, CHINA
Tel (852) 3153 2028 Fax (852) 3153 2074
Website: www.olympiadchampion.com Email: olympiadchampion@gmail.com

香港國際數學競賽總決賽 2025
HONG KONG INTERNATIONAL
MATHEMATICAL OLYMPIAD FINAL 2025

中學二年級 Secondary 2
時限：120 分鐘
Time allowed: 120 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.

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. Given A and B are two non-zero digits and the 2-digit numbers formed by these two digits have the
following properties:
(a) !" is divisible by 9;
(b) The difference between !" and a square number is 1;
Find the 2-digit number !" .
已知 A 和 B 為兩個非零數位。且利用這兩個數位組成的兩位數有以下性質：
(a) !" 可以被 9 整除；
(b) !" 與一個平方數之差是 1；
求兩位數 !" 。

2. Peter goes west for 23km, then goes north for 12km and goes east for 18km. How far is he now from the
original position?
彼得向西走 23 公里，向北走了 12 公里，向東走了 18 公里，問他和原來位置相距多遠？

3. If !"#$%& + !"#$%& = #$%&!" , calculate !"#$%& .

若 !"#$%& + !"#$%& = #$%&!" ，求 !"#$%& 的值。

4. 12th May, 2018 is Saturday, which date of the week will 12th May, 2025 be?
2018 年 5 月 12 日是星期六，之後 2025 年的 5 月 12 日會是星期幾？

5. There are 30 problems in a mathematics competition. The scores of each problem are allocated in the
following ways: 5 marks will be given for a correct answer, 0 marks will be given for a blank answer or a
wrong answer. Find the minimum number of candidate(s) to ensure that 4 candidates will have the same
scores in the competition.
某數學比賽共有 30 條題目。以下述方式為每個題目評分：答對得 5 分、答錯或不作答得零分。求
最小參賽者的數目才可保證比賽中有四人同分。

6. Now the light is on. Amy presses the light button !"#%!"#$ times. Can you guess whether the light is on
or off?
現時燈是亮的，艾美按燈掣 !"#%!"#$ 次，請猜猜現時燈是亮或熄的。

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.

Algebra
代數

7. Find the value of y if ! + !"# + !"$ ! " = " .

若 ! + !"# + !"$ ! " = " ，求 y 的值。

! ! + !" = "
8. Solve " . Find the value of y.
$" ! # " = ##
! ! + !" = "
解" ，求 y 的值。
$" ! # " = ##

9. If x and y are positive integers and !! + " " = !! , find the minimum value of ! + " .
若 x 和 y 是正整數且 !! + " " = !! ，求 ! + " 的最小值。

" + ! + # +…+ !$
10. Evaluate .
$%" + $%!! + $%#! +…+ "!
!

" + ! + # +…+ !$
求 ! 的值。
$%" + $%!! + $%#! +…+ "!

11. Find the value of ""! + "#! + "!! + $$$ + "%! + "&! + #'! .
求 ""! + "#! + "!! + $$$ + "%! + "&! + #'! 的值。

12. Given a and b are positive real numbers and satisfy equations ! ! + " ! = "# and ! + " = ! , find the value
of !" .
已知 a、b 皆為正實數且滿足等式 ! ! + " ! = "# 和 ! + " = ! ，求 !" 的值。

Number Theory
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.

數論

13. A 3-digit number has exactly 3 positive factors. Find the maximum value of this 3-digit number.
一個三位數恰好有 3 個正因數，求這個三位數的最大值。

14. Find the number of all positive factor(s) of the 830.

求 830 的正因數數量。

15. How many simplified fraction(s) with denominator 101 is / are there?
分母為 101 的最簡真分數有多少個？

! " # !$$% !$$& !$$'

+
16. Find the integral part of + +!+ + + .
"$!' "$!' "$!' "$!' "$!' "$!'
! " # !$$% !$$& !$$'
求 + + +!+ + + 的整數部分。
"$!' "$!' "$!' "$!' "$!' "$!'

! " # $ !%
17. Find the decimal part of !
! ! !! ! ! .
"%!& "%!& "%!& "%!& "%!&
! " # $ !%
求! ! ! !! ! ! 的小數部分。
"%!& "%!& "%!& "%!& "%!&

18. It is known that x is rational, ! > ! and ! = ! ! ! … . Find the value of x.

已知有理數 ! > ! 且 ! = ! ! ! … ，求 x 的值。

Geometry
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.

幾何

19. It is known the two sides adjacent to the right angle of a right-angled triangle are in the ratio of !" # . If
the length of the hypotenuse is 16, find the area of the triangle.
已知某直角三角形的兩條直角邊的長度比為 !" # 。若斜邊為 16，求三角形的面積。

20. For four points on a coordinate plane !!"# $%# "!"# &%# # !!'# &%# A , find the area of the rectangle formed
by using those four points as vertices.
直角坐標上有 4 個點分別為 !!"# $%# "!"# &%# # !!'# &%# A ，求以該 4 點為頂點的長方形的面積。

!" !"
21. In the figure below, !" ! #A , !" ! !# and
= ! , evaluate .
!# !#
!" !"
下圖中 !" ! #A 及 !" ! !# 且 = ! ，求 。
!# !#

Question 21
第 21 題

22. What is sum of exterior angles of a decagon (10-sided polygon)?

十邊形外角之和是多少？

23. For rectangles whose side lengths are integers (in cm) and its area is 2015cm2 , how many different
values of perimeter can it take?
一長方形邊長是整數厘米，且其面積是 2015 平方厘米，這個長方形可以有多少個不同的周界長度？

24. Find the value of h in the attached figure. (Show your answer as the simplified fraction if necessary.)
參考附圖，求 h 的值。（如有需要，以最簡分數表示答案）

24
7 h

25
Question 24
第 24 題

Combinatorics
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.

組合數學

25. It is given that the total weight of 4 apples and 3 melons is 1700g and the total weight of 3 apples and 4
melons is 1800g. What is the weight of 1 melon?
已知 4 個蘋果與 3 個蜜瓜共重 1700 克，而 3 個蘋果與 4 個蜜瓜共重 1800 克；那麼 1 個蜜瓜重多
少克？

26. There are 6 boys and 9 girls in a class. Choose 3 boys and 3 girls to form a discussion group. How many
way(s) is / are there in total?
某班有 6 個男生和 9 個女生，從中選取 3 名男生及 3 名女生組成討論小組，問一共有多少個方法？

27. A fair 6-face die is thrown 3 times. Find the probability that the sum of numbers obtained is 17.
擲一枚均質六面骰子三次。求擲得點數之和為 17 的概率。

28. Andy has 29 $10 and $10 banknotes altogether. He has $420 in total. How many $10 banknote(s) does he
have?
安迪有 10 元和 20 元紙幣共 29 張，合共 420 元。問安迪有多少個 10 元紙幣？

29. Inside a 52-card poker, 2 cards are chosen. Find the probability that the sum of numbers obtained is 17.
( ! !""# " !"$# # !"% )
從一副 52 張牌的撲克中抽出 2 張牌。求點數之和為 17 的概率。（ ! !""# " !"$# # !"% ）

30. There are 47 students in Grade 8. 27 and 25 students like playing basketball and volleyball respectively.
However, we don’t know the number of students who like both ball games. Find the maximum number of
students who don’t like both ball games.
八年級有 47 人，其中 27 人喜歡打籃球，25 人喜歡打排球，但不知道兩項皆喜歡的人數，那麼兩
項球類都不喜歡的人最多有多少個？

~ 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.