TIMO Final 2019 卷P5f [PDF]

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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 / 9310 1240

Fax (852) 3153 2074

Website: www.olympiadchampion.com Email: [email protected]

THAILAND

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

小學五年級 Primary 5 時限:120 分鐘 Time allowed:

120

minutes

試題 Question 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.

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) 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.

The age of Samuel 5 years later is equal to the age of Joseph 4 years ago. Given that the age sum between them is 51 this year, how old is Samuel now? 森姆 5 年後的年齡與約瑟夫 4 年前的年齡相等。已知兩人今年年齡和是 51,那麼森姆今年多少歲? 森姆 5 年后的年龄与约瑟夫 4 年前的年龄相等。已知两人今年年龄和是 51,那么森姆今年多少岁?

2.

It requires 30 minutes to cut a piece of wood into 7 sections. If the time required to cut into each section is the same, how many minute(s) is / are required to cut into 14 sections? 一根木材,截成 7 段需要 30 分鐘時間;如果每截一段的時間都相同,那麼把一根木材截成 14 段 需要多少分鐘? 一根木材,截成 7 段需要 30 分钟时间;如果每截一段的时间都相同,那么把一根木材截成 14 段 需要多少分钟?

3.

At a birthday party, if we need to ensure there will be at least 5 people with the same date of birth in Year 1990, how many people are there at the party? 在一個生日會中,若能保證最少有 5 人於 1990 年同一天生日,問這個生日會有多少人? 在一个生日会中,若能保证最少有 5 人于 1990 年同一天生日,问这个生日会有多少人?

4.

It takes A, B, and C 6, 8 and 24 days respectively to finish the job alone. If three people work together, at which day the job is finished? 一項工程,甲單獨要 6 天完成,乙則要 8 天,丙要 24 天,問三人合作,會在第幾天完成這項工程? 一项工程,甲单独要 6 天完成,乙则要 8 天,丙要 24 天,问三人合作,会在第几天完成这项工程?

5.

There are a total of 78 chickens and rabbits in a farm. The animals have a total of 200 legs. How many chicken(s) is / are there? 農場裡有雞和兔共 78 隻,共有 200 條腿。那麼雞有多少隻? 农场里有鸡和兔共 78 只,共有 200 条腿。那么鸡有多少只?

6.

Define the operation symbol: 定義運算如下: 定义运算如下: 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. 2  5  2  3 4  5 3  8  3 4  5  6  7  8

Find the value of (6  10)  (5  9) .

5  9  5 6 7 8 9

求 (6  10)  (5  9) 的值。 求 (6  10)  (5  9) 的值。 Arithmetic 算術 算术 1 1 1 1    ...  . 10 11 1112 12 13 49  50 1 1 1 1    ...  求 的值。 10 11 1112 12 13 49  50 1 1 1 1    ...  求 的值。 10 11 1112 12 13 49  50

7.

Find the value of

8.

Find the value of 112  89  224  21  336  23 . 求 112  89  224  21  336  23 的值。 求 112  89  224  21  336  23 的值。

9.

Find the value of 24062  15942 . 求 24062  15942 的值。 求 24062  15942 的值。

10. Find the value of 800 1  800 10  800  4  800  5 . 求 800 1  800 10  800  4  800  5 的值。 求 800 1  800 10  800  4  800  5 的值。 11. Find the value of 2222  4444  3333111111111111 . 求 2222  4444  3333111111111111 的值。 求 2222  4444  3333111111111111 的值。 12. Find the value of 4.06  2500  40.6 120  406  63 . 求 4.06  2500  40.6 120  406  63 的值。 求 4.06  2500  40.6 120  406  63 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 9-digit number 20190406A is divisible by 16, find the value of A. 若 9 位數 20190406A 可被 16 整除,求 A 的值。 若 9 位数 20190406A 可被 16 整除,求 A 的值。 14. Find the unit digit of A if A  2019  2019  2019  ...  2019  406  406  ... 406 . 2019' s

406' s

若 A  2019  2019  2019  ...  2019  406  406  ... 406 ,求 A 的個位數的值。 2019' s

406' s

若 A  2019  2019  2019  ...  2019  406  406  ... 406 ,求 A 的个位数的值。 2019' s

406' s

15. Find the number of 2-digit prime number(s). 求兩位質數的數目。 求两位质数的数目。 16. The sum of A and B is 406. A is 57 times of B. Find the value of A. A、B 兩數之和是 406,A 是 B 的 57 倍,求 A 的值。 A、B 两数之和是 406,A 是 B 的 57 倍,求 A 的值。 17. What is the largest 4-digit number (without repeated digits) that can be divisible by 12 and 16? 求最大的四位數且數位沒有重複能同時被 12 及 16 整除。 求最大的四位数且数字没有重复能同时被 12 及 16 整除。 18. The sum of nine consecutive even numbers is 406. Find the largest number. 9 個連續偶數之和為 406,求最大的數。 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. 9 个连续偶数之和为 406,求最大的数。

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 area of a rectangle is 406. 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 406,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 406,边长为整数,矩形的周界有多少个不同可能值? 21. Now there are 8 points. How many triangle(s) can be formed? 現有 8 點,可組成多少個三角形? 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. 现有 8 点,可组成多少个三角形? 22. 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? 请问下图有多少个含有「*」的矩形?

*

Question 22 第 22 題 第 22 题 23. It is given that the area of a rectangle is 196. The width and length are integers. Find the smallest value of the perimeter. 已知一個矩形面積為 196,而長和闊均是整數,求周界的最小值。 已知一个矩形面积为 196,而长和阔均是整数,求周界的最小值。 24. It is known as the lengths of shortest and the longest sides for a right-angled triangle are 25 and 65 respectively. Find the area of the triangle. 已知一個直角三角形最短及最長條邊分別為 25 及 65,求三角形的面積。 已知一个直角三角形最短及最长条边分别为 25 及 65,求三角形的面积。

Combinatorics 組合數學 组合数学 25. There are 5 students passing the balls in an order. Andy holds the ball first. After 5-time pass, Andy still holds the ball now. How many way(s) is / are there? 現時 5 名學生在傳球,首先球在安迪手上,經過 5 次傳送後,球仍在安迪手上,共有多少種走法? 现时 5 名学生在传球,首先球在安迪手上,经过 5 次传送后,球仍在安迪手上,共有多少种走法? 26. How many 2-digit number(s) is / are there such that the sum of the digits is divisible by 5 and no repeated digits are used? 有多少個數位之和是 5 的倍數且沒有重複數位的兩位數? 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. 有多少个数字之和是 5 的倍数且没有重复数字的两位数? 27. Choose 3 digits, without repetition, from 2, 4, 6, 8, 9 to form 3-digit numbers. How many number(s) can be divisible by 8? 從 2、4、6、8、9 中選 3 個不可重複的數位組成三位數。請問當中有多少個數能被 8 整除? 从 2、4、6、8、9 中选 3 个不可重复的数字组成三位數。请问当中有多少个数能被 8 整除? 28. Numbers are drawn from 150 integers 1 to 150. At least how many number(s) is / are drawn at random to ensure that there are two numbers whose difference is 30? 在 1 至 150 這 150 個整數中最少任意選出多少個數,才必定有兩個數之差是 30? 在 1 至 150 这 150 个整数中最少任意选出多少个数,才必定有两个数之差是 30? 29. If Alice goes from point A to point B, each step can only move up or move right. How many way(s) is / are there? 如果愛麗絲從 A 點走到 B 點,每步只能向上或向右走,共有多少種走法? 如果艾丽斯从 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. According to the following sequence for first 2019th terms, how many odd number(s) is / are there? 求下列數列中,首 2019 項中有多少個奇數?(奇數即是單數) 求下列数列中,首 2019 项中有多少个奇数?(奇数即是单数) 1, 1, 2, 3, 5, 8, 13, 21, …

~ 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.