TIMO Mock 2019 卷P4f [PDF]

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泰國國際數學競賽總決賽 2019 INTERNATIONAL MATHEMATICAL OLYMPIAD FINAL 2019

小學四年級 Primary 4 時限：90 分鐘 Time allowed:

90

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. 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. 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. 填空題（第 1 至 30 題）（每題 5 分，答錯及空題不扣分） 填空题（第 1 至 30 题）（每题 5 分，答错及空题不扣分） Open-Ended Questions (1st ~30th) (5 points for correct answer, no penalty point for wrong answer) Logical Thinking 邏輯思維 逻辑思维 1.

What is the value of the number to represent “?” in the following sequence? 求在以下數列中？代表數字的值。 求在以下数列中？代表数字的值。 1, 1, 1, 3, 5, 9, 17, 31, ？ , …

2.

According to the pattern shown below, what is the 203rd symbol counting from the left? 按以下規律，由左數起，由第 1 個至第 203 個符號共有多少個圓形？ 按以下规律，由左数起，由第 1 个至第 203 个符号共有多少个圆形？

○ △ □ ○ ○ △ □ ○ ○ △ □ ○… 3.

The age of Samuel 7 years later is equal to the age of Joseph 3 years later. Given that Joseph is 42 years old now, how old is Samuel now? 森姆 7 年後的年齡與約瑟夫 3 年後的年齡相等。已知約瑟夫今年 42 歲，那麼森姆今年多少歲？ 森姆 7 年后的年龄与约瑟夫 3 年后的年龄相等。已知约瑟夫今年 42 岁，那么森姆今年多少岁？

4.

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。 按以下规律，求在横在线的数字。 1 、 8 、 27 、 64 、 __ 、….

5.

There are a total of 51 tigers and cranes in a zoo. The animals have a total of 114 legs. How many tiger(s) is / are there? 動物園裡有老虎和鶴共 51 隻，共有 114 條腿。那麼老虎有多少隻？ 动物园里有老虎和鹤共 51 只，共有 114 条腿。那么老虎有多少只？ 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.

6.

Today is 5th April, Friday. Which day of the week is 31st August? 今天是 4 月 5 日星期五，8 月 31 日是星期幾？ 今天是 4 月 5 日星期五，8 月 31 日是星期几？

Arithmetic 算術 算术 7.

Find the value of 90 1  90 10  90  5  90  2 . 求 90 1  90 10  90  5  90  2 的值。 求 90 1  90 10  90  5  90  2 的值。

8.

Find the value of 1  5  9  ...  81  85  89 . 求 1  5  9  ...  81  85  89 的值。 求 1  5  9  ...  81  85  89 的值。

9.

Find the value of 1  2  3  ... 19  20 19  ...  3  2 . 求 1  2  3  ... 19  20 19  ...  3  2 的值。 求 1  2  3  ... 19  20 19  ...  3  2 的值。

10. Find the value of 1  4  7 10 13 16 19  22  25  28  31  34  37  40  43 . 求 1  4  7 10 13 16 19  22  25  28  31  34  37  40  43 的值。 求 1  4  7 10 13 16 19  22  25  28  31  34  37  40  43 的值。 11. Find the value of 272 12 15 272  272  23 . 求 272 12 15 272  272  23 的值。 求 272 12 15 272  272  23 的值。 12. Find the value of 3125  32 16 125 求 3125  32 16 125 的值。 求 3125  32 16 125 的值。 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.

Number Theory 數論 数论 13. If a 8-digit number 20190405A is divisible by 9, find the value of A. 若 8 位數 20190405A 可被 9 整除，求 A 的值。 若 8 位数 20190405A 可被 9 整除，求 A 的值。 14. Find the unit digit of A if A  5  5  5  ...  5  7  7  7  ...  7 . 2019' s

2019' s

若 A  5  5  5  ...  5  7  7  7  ...  7 ，求 A 的個位數的值。 2019' s

2019' s

若 A  5  5  5  ...  5  7  7  7  ...  7 ，求 A 的个位数的值。 2019' s

2019' s

15. 6 students have 60 balloons in total and each of them has a different number of balloons and at least 1 balloon. At most how many balloon(s) does the student with the most balloons have? 6 個小孩一共有 60 個氣球。已知當中沒有任何小孩有相同數量的氣球，而每人至少有 1 個，問擁 有最多氣球的小孩最多擁有多少個氣球？ 6 个小孩一共有 60 个气球。已知当中没有任何小孩有相同数量的气球，而每人至少有 1 个，问拥 有最多气球的小孩最多拥有多少个气球？ 16. The sum of seven consecutive numbers is 294. Find the smallest number. 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. 7 個連續數之和為 294，求最小的數。 7 个连续数之和为 294，求最小的数。 17. What is the smallest 4-digit number that can be divisible by 8 and 12? 求最小的四位數能同時被 8 及 12 整除。 求最小的四位数能同时被 8 及 12 整除。 18. Define the operation symbol a  b  a  b  a , find the value of (12  4)  3 . 定義運算 a  b  a  b  a ，求 (12  4)  3 的值。 定义运算 a  b  a  b  a ，求 (12  4)  3 的值。

Geometry 幾何 几何 19. How many rectangle(s) is / are there in the figure below? 請問下圖有多少個長方形？ 请问下图有多少个长方形？

Question 19 第 19 題 第 19 题 20. The perimeter of a rectangle is 100. Find the maximum value of the area of this rectangle. 矩形的周界是 100，求矩形面積的最大值。 矩形的周界是 100，求矩形面积的最大值。 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.

21.

The area of a rectangle is 60. If the sides of the rectangle are integers, how many different value(s) of the perimeter of this rectangle is / are there? 一個矩形的面積為 60，邊長為整數，矩形的周界有多少個不同可能值？ 一个矩形的面积为 60，边长为整数，矩形的周界有多少个不同可能值？

22. There are 7 points on a plane. How many quadrilateral(s) can you draw? 在平面上有 7 點，可以畫出多少個四邊形？ 在平面上有 7 点，可以画出多少个四边形？ 23. It is known as the lengths of shorter sides for a right-angled triangle are 6 and 8 respectively. Find the length of the longest length. 已知一個直角三角形較短的兩條邊分別為 6 及 8，求最長邊長的長度。 已知一个直角三角形较短的两条边分别为 6 及 8，求最长边长的长度。 24. Now there are 19 points. How many line segment(s) can be formed? 現有 19 點，共可組成多少條線段？ 现有 19 点，共可组成多少条线段？

Combinatorics 組合數學 组合数学 25. A flight of stairs has 6 steps. David can go up for 1 step, 2 steps or 3 steps each time. How many way(s) is / are there for David to go up the stairs? 一道有 6 級的樓梯，大衛每一步可以上 1 級、2 級或 3 級，問大衛走上這道樓梯共有多少種方法？ 一道有 6 级的楼梯，戴维每一步可以上 1 级、2 级或 3 级，问戴维走上这道楼梯共有多少种方法？ 26. According to the following sequence for first 399th terms, how many odd number(s) is / are there? 求下列數列中，首 399 項中有多少個奇數？（奇數即是單數） 求下列数列中，首 399 项中有多少个奇数？（奇数即是单数） 1, 2, 3, 6, 11, 20, 37, 68, … 27. Choose 2 digits, without repetition, from 1, 3, 5, 6, 7 to form 2-digit numbers. How many number(s) can 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.

be divisible by 3? 從 1、3、5、6、7 中選 2 個不可重複的數位組成兩位數。請問當中有多少個數能被 3 整除？ 从 1、3、5、6、7 中选 2 个不可重复的数字组成两位數。请问当中有多少个数能被 3 整除？ 28. How many 2-digit number(s) is / are there such that the number contains digit “7” and no digits are repeated? 有多少個包含數位 7 且沒有重複數位的兩位數？ 有多少个包含数字 7 且没有重复数字的两位数？ 29. If Andy goes from point A to point B, each step can only move up or move right. How many way(s) is / are there? 如果安迪從 A 點走到 B 點，每步只能向上或向右走，共有多少種走法？ 如果安迪从 A 点走到 B 点，每步只能向上或向右走，共有多少种走法？ B

A Question 29 第 29 題 第 29 题 30. Numbers are drawn from 50 integers 1 to 50. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose sum is 23? 在 1 至 50 這 50 個整數中最少任意選出多少個數，才必定有兩個數之和是 23？ 在 1 至 50 这 50 个整数中最少任意选出多少个数，才必定有两个数之和是 23？ ~ 全卷完 ~ ~ 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.