Timo Contest

Thailand International
Mathematical Olympiad 2015

泰國國際數學競賽 2015
Secondary 1 Past Paper Booklet
中學一年級試題集
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 5 題，共 25 題，每題 4 分，總分 100 分，答錯不扣分。
There are 5 exam areas and 5 questions in each exam area. There are a total of 25
questions in this Question-Answer Book. Each carries 4 marks. Total score is 100
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.
泰國國際數學競賽 2015 中學一年級試題集
Thailand International Mathematical Olympiad 2015 Secondary 1 Past Paper Booklet
填空題（第 1 至 25 題）（每題 4 分，答錯及空題不扣分）

Open-Ended Questions (1st ~25th) (4 points for correct answer, no penalty point for wrong answer)
Logical Thinking
邏輯思維
1. Amy bought flowers from a market. It is known that a rose, a lily and a tulip cost $16, $8 and $1
respectively. She used $300 to buy 30 flowers and bought at least 1 for each type. How many roses did
she buy?
艾美到市場買花。已知每朵玫瑰、每朵百合和每朵鬱金香的售價分別為 16 元、8 元和 1 元。她用
了 300 元買了 30 朵花，而且每種花都最少買一朵。她一共買了多少朵玫瑰？
2. According to the pattern shown below, what is the next number?

按以下規律，問下一個數是甚麼？
1、2、3、6、11、20、37、68、…
3. Find the integral part of .
求的整數部分。
4. There are six teams named A, B, C, D, E and F, participating a tournament. In 5 days, each team will play
one game in each day. They play with another team once in the tournament. So there are 3 tournaments
every day. Given that:
1) Team A wins Team B on the first day.
2) Team C is defeated by Team D on the second day.
3) Team E wins Team A on the third day.
4) Team B wins Team C on the fourth day.
Which team does Team A play with on the fifth day？
有 A、B、C、D、E、F 六個籃球隊參加循環賽，賽程總共五天，每隊每天出賽一次，五天裡各與
不同的隊伍比賽一場，所以每天都有三場比賽，已知：
1) 第一天 A 隊贏了 B 隊
2) 第二天 C 隊輸給了 D 隊
3) 第三天 E 隊贏了 A 隊
4) 第四天 B 隊贏了 C 隊
請問，A 隊在第五天與哪一隊比賽？
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.
5. There are 7 people arguing which day of week is today. Their statements are quoted as below:
A: Yesterday was Wednesday.
B: Tomorrow will be Tuesday.
C: Tomorrow will be Wednesday.
D: The day after tomorrow will be Tuesday.
E: Today is Tuesday.
F: Today is neither Monday, Tuesday nor Sunday.
G: Today is not Saturday.
Given that among them only one statement is correct, which day of week is today?
有七個人在爭論今天是星期幾，他們的說法如下：
A：昨天是星期三。
B：明天是星期二。
C：明天是星期三。
D：後天是星期二。
E：今天是星期二。
F：今天不是星期一，也不是星期二，也不是星期天。
G：今天不是星期六。
已知七個人當中只有一個人說對了，那麼今天是星期幾？
Algebra
代數
6. Given a is a real number and , find the sum of all possible value of a.
已知 a 是一個實數且，求 a 的所有可能值之和。
7. Find the value of .
求的值。
8. Convert into recurring decimals.
把轉化為循環小數。
9. Given a is a positive integer and , how many possible values for a?

已知 a 是一個正整數且，a 有多少個可能值？
求的值。
Number Theory
數論
11. If n is a positive integer and are all prime numbers, find the value of n.
若 n 為正整數且皆為質數，求 n 的值。
12. Find the remainder when is divided by 8.

求除以 8 所得的餘數。
13. Given x and y are natural numbers and . When y attains minimum, find the value of x.
已知皆為自然數且。求 x 的值使 y 的數值為最小。
14. How many simplest fractions whose denominator is 2015 are there?
請問有多少個分母為 2015 的最簡真分數？
15. Given that positive integers x and y ( ) satisfy equations and . Find the
value of .
已知正整數 x,y ( ) 滿足算式及，求的值。
Geometry
幾何
16. Combine 2015 equilateral triangles with sides 1 unit to form a quadrilateral. Find the maximum perimeter
of that quadrilateral.
把 2015 個邊長為 1 的等邊三角形拼合成一個四邊形，求該四邊形周界的最大值。
17. Shown in the graph, ABCDEFGH is a cube. If the distance between of A and F is , find the volume
of the cube.
如圖所示，ABCDEFGH 為一立方體。若 A 點與 F 點之間的距離為。求立方體的體積。
Question 17
第 17 題
18. In the diagram, ABCD is a square of side length 20 units. If , find the length of
AG.
圖中 ABCD 是一個邊長為 20 單位的正方形。若，求 AG 的長度。
Question 18
第 18 題
19. A circle is inscribed in a square and the circle also inscribes another small square. If the side length of the
larger square is 4cm, find the area of the smaller square in cm2.
已知一個正方形內接一個圓形，而這個圓形亦內接另一個小正方形。若大正方形的邊長為 4 厘米，
求小正方形的面積（以平方厘米表示答案）。
20. 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 10cm, find the area of the triangle in cm 2.
已知某直角三角形的兩條直角邊的長度比為。若斜邊為 10 厘米，求三角形的面積（以平方厘
米表示答案）。
Combinatorics
組合數學
21. A collection of integers chosen from 1 to 2015 has the property that none of its members are 10 times of
another. What is the maximum number of members such a collection can have?
由 1 到 2015 當中抽取一組整數，使得當中沒有任何一個數字是另外一個的 10 倍。問該組最多可
以有多少個數字？
22. Basketball team has 15 members. How many different ways are there to send 10 teammates to participate
in a competition in which one of them is the captain?
籃球隊有 15 人，從中選取 10 人參加比賽，其中一名為隊長，問一共有多少種方法？
23. Given are all positive integers, how many solutions are there to the inequality
?
已知皆為正整數，問不等式有多少個解組？
24. Five students, A, B, C, D and E, are passing a ball, initially A holds the ball. Given A passes the ball to D
at 5th pass, how many possible ways of passing the ball are there?
A、B、C、D 和 E 5 個同學互相傳球，開始時球在 A 同學手中。已知第 5 次傳球時， A 同學傳球
給 D 同學，問有多少個可能的傳球路徑？
25. Choosing from numbers 0,2,4,6 and 8 as digits, how many positive integers less than 10000 can be
formed? (Each number cannot be used more than once)
從數字 0、2、4、6 和 8 當中選取數字作為數位，可以組成多少個少於 10000 的正整數？
﹝數字不可以重覆﹞
~ 全卷完 ~
~ End of Paper ~
Solutions
題解
Logical Thinking
邏輯思維
1. Amy bought flowers from a market. It is known that a rose, a lily and a tulip cost $16, $8 and $1
respectively. She used $300 to buy 30 flowers and bought at least 1 for each type. How many roses did
she buy?
艾美到市場買花。已知每朵玫瑰、每朵百合和每朵鬱金香的售價分別為 16 元、8 元和 1 元。她用
了 300 元買了 30 朵花，而且每種花都最少買一朵。她一共買了多少朵玫瑰？
解：11
Let the number of roses, lilies and tulips are x, y and z respectively.
We have equations .
By elimination, we have .
As y should be integer, we have should be multiple of 7.
So possible values of x are .
Since z and y should be positive integers, 。

The number of roses, lilies and tulips are 11, 15 and 4 respectively
設玫瑰的數目為 x，百合的數量為 y 及鬱金香的數量為 z。
可得聯立方程
利用消元法，可得
由於 y 必須是整數，故此一定是 7 的倍數。
可得的可能值為。
考慮 z 及 y 皆為正整數，所以。故此每種花的購買數目只可以是「11 朵玫瑰，15 朵百合及 4

朵鬱金香」。
2. According to the pattern shown below, what is the next number?

按以下規律，問下一個數是甚麼？
1、2、3、6、11、20、37、68、…
解：125
From the fourth term, each term is the sum of 3 previous terms.
由第四項起，每一項皆為前三項之和。
3. Find the integral part of .
求的整數部分。
解：1
The integral part of is 1.
的整數部份是 1。
4. There are six teams named A, B, C, D, E and F, participating a tournament. In 5 days, each team will play
one game in each day. They play with another team once in the tournament. Given that:
1) Team A wins Team B on the first day.
2) Team C is defeated by Team D on the second day.
3) Team E wins Team A on the third day.
4) Team B wins Team C on the third day.
Which team does Team A play with on the fifth day？
有 A、B、C、D、E、F 六個籃球隊參加循環賽，賽程總共五天，每隊每天出賽一次，五天裡各與
不同的隊伍比賽一場，所以每天都有三場比賽，已知：
1) 第一天 A 隊贏了 B 隊
2) 第二天 C 隊輸給了 D 隊
3) 第三天 E 隊贏了 A 隊
4) 第四天 B 隊贏了 C 隊
請問，A 隊在第五天與哪一隊比賽？
解：Team C / C 隊
Team A played with Team B and Team E on the first day and the third day respectively.
Among Teams C, D and F, as Team C can play with Team A on neither second day nor fourth day, Team
C should play with Team A on the fifth day.
A 隊分別與 B 隊及 E 隊於第一天和第三天對賽。
在 C 隊、D 隊及 F 隊之中，由於 C 隊不能在第二天或第四天與 A 隊對賽，故 C 隊只能在第五天與
A 隊對賽。
5. There are 7 people arguing which day of week is today. Their statements are quoted as below:
A: Yesterday was Wednesday.
B: Tomorrow will be Tuesday.
C: Tomorrow will be Wednesday.
D: The day after tomorrow will be Tuesday.
E: Today is Tuesday.
F: Today is neither Monday, Tuesday nor Sunday.
G: Today is not Saturday.
Given that among them only one statement is correct, which day of week is today?
有七個人在爭論今天是星期幾，他們的說法如下：
A：昨天是星期三。
B：明天是星期二。
C：明天是星期三。
D：後天是星期二。
E：今天是星期二。
F：今天不是星期一，也不是星期二，也不是星期天。
G：今天不是星期六。
已知七個人當中只有一個人說對了，那麼今天是星期幾？
解：Saturday / 星期六
Converting their statements, we will have the following table:
經過轉換他們的說法可得出下表：
A B C D E F G
Monday
星期一
Tuesday
星期二
Wednesday
星期三
Thursday
星期四
Friday
星期五
Saturday
星期六
Sunday
星期日
Among them only Saturday satisfies the condition that “only one statement is correct”.
當中只有星期六符合「只有一個人說對了」這個條件。
Algebra
代數
6. Given a is a real number and , find the sum of all possible value of a.
已知 a 是一個實數且，求 a 的所有可能值之和。
解：
(rejected / 捨去)
The sum of all possible value is .

所有可能值之和為。
求的值。
解：4,290
8. Convert into recurring decimals.
把轉化為循環小數。
解：
9. Given a is a positive integer and , how many possible values of a?

已知 a 是一個正整數且，a 有多少個可能值？
解：6
Possible values of a: 1, 2, 3, 4, 5 and 6, 6 in total.

a 的可能值: 1、2、3、4、5 及 6，共 6 個。
求的值。
解：
Number Theory
數論
11. If n is a positive integer and are all prime numbers, find the value of n.
若 n 為正整數且皆為質數，求 n 的值。
解：9
Since one of and should be 2, we have .
The prime numbers are , and .
考慮到及之間必有一個數字為 2，可得。
由此可得三個質數為、及。
12. Find the remainder when is divided by 8.

求除以 8 所得的餘數。
解：7
The remainder of is 7.
The remainder of is 1.
So we conclude that the pattern of the remainder of will be .
Since 2015 is an odd number, the remainder will be 7.
2015 除以 8 的餘數為 7。
除以 8 的餘數為 1。
由此可得除以 8 的餘數的周期為。
由於 2015 為單數，所以餘數為 7。
13. Given x and y are natural numbers and . When y attains minimum, find the value of x.
已知皆為自然數且。求 x 的值使 y 的數值為最小。
解：3
14. How many simplest fractions whose denominator is 2015 are there?
請問有多少個分母為 2015 的最簡真分數？
解：1,440
Consider , so among the numerators 1–2015 there are can be simplified by 5, can
be simplified by 13 and can be simplified by 31.
So the answer is
考慮，所以分子為 1 至 2015 中有可被 5 約簡、有可被 13 約簡、有可被
31 約簡。因此答案為
15. Given that positive integers x and y ( ) satisfy equations and . Find the
value of .
已知正整數 x,y ( ) 滿足算式及，求的值。
解：25
Only and satisfy the two equations, so .

只有及滿足上述算式，所以。
Geometry
幾何
16. Combine 2015 equilateral triangles with sides 1 unit to form a quadrilateral. Find the maximum perimeter
of that quadrilateral.
把 2015 個邊長為 1 的等邊三角形拼合成一個四邊形，求該四邊形周界的最大值。
解：4,031
The perimeter of quadrilateral attains maximum if all the equilateral triangles are arranged in a row.
The maximum value of perimeter is .
當所有等邊三角形排成一條橫行時，所得的周界為最大。
所以四邊形周界最大值為。
17. Shown in the graph, ABCDEFGH is a cube. If the distance between of A and F is , find the volume
of the cube.
如圖所示，ABCDEFGH 為一立方體。若 A 點與 F 點之間的距離為。求立方體的體積。
Question 17
第 17 題
解：64
Let the side length of the cube be x.
By Pythagoras Theorem we have

So and the volume is 64.
設正方體邊長為 x
利用畢氏定理可得
所以，體積等於 64。
18. In the diagram, ABCD is a square of side length 20 units. If , find the length of
AG.
圖中 ABCD 是一個邊長為 20 單位的正方形。若，求 AG 的長度。
Question 18
第 18 題
解：
Add straight lines one the graph, as shown in the figure below.
Consider the area of the square we have .
By . Pythagoras Theorem we have .

在正方形上分別加上四條直線，得出如下圖的圖案。
可得
利用畢氏定理，可得
19. A circle is inscribed in a square and the circle also inscribes another small square. If the side length of the
larger square is 4cm, find the area of the smaller square.
已知一個正方形內接一個圓形，而這個圓形亦內接另一個小正方形。若大正方形的邊長為 4 厘米，
求小正方形的面積。
解：8
The area of smaller square is half of the bigger square.
So the area of smaller square is cm2.
小正方形剛好是大正方形的面積的一半。
小正方形的面積厘米。
20. 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 10cm, find the area of the triangle in cm .2
已知某直角三角形的兩條直角邊的長度比為。若斜邊為 10 厘米，求三角形的面積 (以平方厘

米表示答案) 。
解：7
Let the length of the shorter side adjacent to the right angle of the triangle be x, so that of the longer side
will be 7x.
By Pythagoras Theorem, we have .
The area of the triangle is equal to .

設較短的直角邊的邊長為 x，可得較長直角邊的邊長為 7x。
由畢氏定理可得。
由此三角形面積為。
Combinatorics
組合數學
21. A collection of integers chosen from 1 to 2015 has the property that none of its members are 10 times of
another. What is the maximum number of members such a collection can have?
由 1 到 2015 當中抽取一組整數，使得當中沒有任何一個數字是另外一個的 10 倍。問該組最多可
以有多少個數字？
解：1,832
Following the question the answer can be found by counting numbers in the form of (n is an even
number and k is not multiple of 10).
按題意即為找出由 1 至 2015 內以形式﹝n 為偶數，k 為不是 10 的倍數的整數﹞的數字。
22. Basketball team has 15 members. How many different ways are there to send 10 teammates to participate
in a competition in which one of them is the captain?
籃球隊有 15 人，從中選取 10 人參加比賽，其中一名為隊長，問一共有多少種方法？
解：30,030
There are methods of choosing teammates and for each combination there are 10 choices for
captain. So the answer is .
一共有個選拔隊員的組合，對於每個組合都有 10 個選擇隊長的情況。所以答案為
。
23. Given are all positive integers, how many solutions are there to the inequality
?
已知皆為正整數，問不等式有多少個解組？
解：126
Inequality is equivalent to the equation where is also a
positive integer. So the number of solutions is .
不等式等價於等式且是正整數。所以解組數目為
。
24. Five students, A, B, C, D and E, are passing a ball, initially A holds the ball. Given A passes the ball to D
at 5th pass, how many possible ways of passing the ball are there?
A、B、C、D 和 E 5 個同學互相傳球，開始時球在 A 同學手中。已知第 5 次傳球時， A 同學傳球
給 D 同學，問有多少個可能的傳球路徑？
解：52
The number of possible routes of passing is shown as the table below.
傳球路徑的數目如下表所示。
Start After 1st Pass After 2nd Pass After 3rd Pass After 4th Pass After 5th Pass
開始第一次傳球第二次傳球第三次傳球第四次傳球第五次傳球
A A 4 A 12 A 52
B 1 B 3 B 13 B 51
A 1 C 1 C 3 C 13 C 51 D 52
D 1 D 3 D 13 D 51
E 1 E 3 E 13 E 51
25. Choosing from numbers 0,2,4,6 and 8 as digits, how many positive integers less than 10000 can be
formed? (Each number cannot be used more than once)
從數字 0、2、4、6 和 8 當中選取數字作為數位，可以組成多少個少於 10000 的正整數？
﹝數字不可以重覆﹞
解：164
4-digit number:
3-digit number:
2-digit number:
1-digit number:
There are 164 numbers in total.
一共有個四位數、個三位數、個兩位數及 4 個一位數。
合共 164 個數字。
~ 全卷完 ~
~ End of Paper ~
泰國國際數學競賽 2015
THAILAND INTERNATIONAL
MATHEMATICAL OLYMPIAD 2015
中學一年級答案
Secondary 1 Answer Key
Question No Answer Question No Answer Question No Answer
題號答案題號答案題號答案
1 11 11 9 21 1,832
2 125 12 7 22 30,030
3 1 13 3 23 126
Team C /
4 14 1,440 24 52
C隊
Saturday /
5 15 25 25 164
星期六
6 16 4,031
7 4,290 17 64
8 18
9 6 19 8
10 20 7

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Thailand International

Mathematical Olympiad 2015

填空題（第 1 至 25 題）（每題 4 分，答錯及空題不扣分）

2. According to the pattern shown below, what is the next number?

3. Find the integral part of .

7. Find the value of .

8. Convert into recurring decimals.

9. Given a is a positive integer and , how many possible values for a?

10. Find the value of .

12. Find the remainder when is divided by 8.

已知 皆為自然數且 。求 x 的值使 y 的數值為最小。

As y should be integer, we have should be multiple of 7.

So possible values of x are .

Since z and y should be positive integers, 。

由於 y 必須是整數，故此 一定是 7 的倍數。

考慮 z 及 y 皆為正整數，所以 。故此每種花的購買數目只可以是「11 朵玫瑰，15 朵百合及 4

2. According to the pattern shown below, what is the next number?

3. Find the integral part of .

The integral part of is 1.

The sum of all possible value is .

7. Find the value of .

8. Convert into recurring decimals.

9. Given a is a positive integer and , how many possible values of a?

Possible values of a: 1, 2, 3, 4, 5 and 6, 6 in total.

10. Find the value of .

12. Find the remainder when is divided by 8.

已知 皆為自然數且 。求 x 的值使 y 的數值為最小。

be simplified by 13 and can be simplified by 31.

考慮 ，所以分子為 1 至 2015 中有 可被 5 約簡、有 可被 13 約簡、有 可被

Only and satisfy the two equations, so .

By Pythagoras Theorem we have

Consider the area of the square we have .

By . Pythagoras Theorem we have .

已知某直角三角形的兩條直角邊的長度比為 。若斜邊為 10 厘米，求三角形的面積 (以平方厘

By Pythagoras Theorem, we have .

The area of the triangle is equal to .

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已知皆為自然數且。求 x 的值使 y 的數值為最小。

由於 y 必須是整數，故此一定是 7 的倍數。

考慮 z 及 y 皆為正整數，所以。故此每種花的購買數目只可以是「11 朵玫瑰，15 朵百合及 4

已知皆為自然數且。求 x 的值使 y 的數值為最小。

考慮，所以分子為 1 至 2015 中有可被 5 約簡、有可被 13 約簡、有可被

已知某直角三角形的兩條直角邊的長度比為。若斜邊為 10 厘米，求三角形的面積 (以平方厘