Aimo 2019 Trial g4 [PDF]

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2019 Asia International Mathematical Olympiad Open Contest Trial

2019 亞洲國際數學奧林匹克公開賽初賽

Grade 4 – Question Book 小學四年級—問題簿

時限：90 分鐘 Time allowed:

90

minutes

試題 Question Paper

本試題不可取走。 THIS QUESTION PAPER 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 Paper without approval of the examiner. Otherwise, contestant may be DISQUALIFIED.

請將答案寫在 答題紙 上。 All answers should be written on the ANSWER SHEET. 甲部：每題 4 分 Section A – each question carries 4 marks 1)

求以下算式的值。 Find the value of the following expression. 2019  7  2018  9  2005  7  2000  9

2)

已知 10001  73 137 ，求 20192019 最大的三個因數之和。 Given 10001  73 137 , find the sum of the three greatest factors of 20192019.

3)

布袋裏有紅色波子 4 粒、黃色波子 6 粒及藍色波子 5 粒，現在隨意在袋中抽出多少粒波子，才能 保證抽出 2 粒藍色波子？ There are 4 red marbles, 6 yellow marbles and 5 blue marbles in a bag. How many marbles should be picked up randomly to ensure 2 blue marbles are picked?

4)

小匹買 4 盒杯麵和 7 瓶汽水，共付 133 元。已知每盒杯麵的價錢是每瓶汽水價錢的 3 倍，那麼每 盒杯麵的價錢是多少元？ Patrick paid $133 for buying 4 cup noodles and 7 bottles of soft drink. Given that the cost of each cup noodle is 3 times of the cost of each bottle of soft drink. How many dollars is the price of each cup noodle?

5)

如果三角形的底和高都是整數厘米，而三角形面積為 219 平方厘米，那麼有多少個底和高的可能 性？ Given that both the base and the height of a triangle are integers in centimetres and the area of the triangle is 219 square centimetres. How many possible combination(s) of the base and the height is / are there?

6)

有一個數。若把它乘以 5，然後加上 4，便會得出 74。這個數是多少？ There is a number. If it is multiplied by 5, and then added with 4, then the result will be 74. What is this number?

7)

已知一矩形的長和闊都是整數厘米，且周界為 210 厘米，那麼該矩形的最大面積是多少平方厘米？ Given that both the length and the width of a rectangle are integers in centimetres. And the perimeter of the rectangle is 210 centimetres. How many square centimetres is the maximum area of the rectangle?

8)

求以下算式的值。 Find the value of the following expression.

20192  19812 ~ 甲部完 ~ ~ End of Section A ~ 請以最簡形式填寫答案。若計算結果是分數，請化至最簡，並確保為真分數或帶分數，或將計算結果寫成小數。 答案可以根式表示，唯該根式必須是最簡形式。除特別註明外，毋需填寫單位。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. You may use square root to represent the answer which is in the simplest form. Unless otherwise stated, no need to write down any unit. 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. 乙部：每題 5 分 Section B – each question carries 5 marks 9)

求以下算式的值。 Find the value of the following expression. 1 2 3 670 671 672       2019 2019 2019 2019 2019 2019

10) 在圖 10 中，5 個完全一樣的長方形組成一個大長方形，若大長方形的面積是 1470 平方厘米，那麼 一個小長方形的周界是多少厘米？ In Figure 10, a large rectangle is formed by 5 identical small rectangles. Given that the area of the large rectangle is 1470 square centimetres, how many centimetres is the perimeter of a small rectangle?

圖 10 Figure 10 11) 若九位數 2019 AA324 可被 7 整除，求 A 的值。 If the nine-digit number 2019 AA324 is divisible by 7, find the value of A. 12) 有一個周界為 400 米的湖，住在湖邊的小匹打算圍繞湖的旁邊種樹。第一天，他由他的家開始每 隔 4 米便種下一棵蘋果樹，第二天，他由他的家開始每隔 10 米且沒有蘋果樹的地方種下一棵橙樹。 問最後他合共種了多少棵樹？ There is a big lake with perimeter of 400 metres. Patrick, who lives beside the lake, wanted to plant trees around the lake. On the first day, starting from his home, he planted an apple tree every 4 metres. On the second day, starting from his home, he planted an orange tree every 10 metres if there is no apple trees at that position. How many trees has he planted in total? 13) 小奧有一堆 1 元、2 元和 5 元硬幣，已知 1 元硬幣的數目是 2 元硬幣的數目的 14 倍，且這堆硬幣 的總面值是 120 元。求 5 元硬幣的數目。 Oliver has some 1 dollar coins, 2 dollar coins and 5 dollar coins. Given that the number of 1 dollar coins is 14 times of that of 2 dollar coins, and the total value of the coins is 120 dollars. Find the number of 5 dollar coins. 14) 求以下算式的值。 Find the value of the following expression. 10002  9982  9962  9942  9922   82  62  42  22

(C) 2019 Asia International Mathematical Olympiad Union (AIMO) http://aimo.world/ All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic mechanical, photocopying, recording or otherwise, without the prior permission of Asia International Mathematical Olympiad Union.

請將答案寫在 答題紙 上。 All answers should be written on the ANSWER SHEET. 15) 小艾、小貝、小克、小迪四個小朋友共有糖果 200 顆，小艾首先把 13 顆給了小貝，然後小克給了 小迪 20 顆，小貝又給小克她手上所有糖果的一半，最後小迪給了小艾 24 顆後四人有的糖果數目 便相等。那麼小貝原有多少顆糖果？ Amy, Bella, Chris and Debby have 200 candies in total. Amy gave Bella 13 candies. Then Chris gave Debby 20 candies. Then Bella gave Chris half of all her candies. Finally Debby gave Amy 24 candies. Then they have the same number of candies. How many candies did Bella originally have? 16) 四個朋友分別有一些金幣，小艾和小貝共有 32 枚金幣，小貝和小克共有 46 枚金幣，小克和小迪 共有 45 枚金幣。問小艾和小迪共有多少枚金幣？ Four friends have some golden coins. Aiden and Beck have 32 coins together. Beck and Chris have 46 coins together. Chris and Dylan have 45 coins together. How many coins do Aiden and Dylan have together?

~ 乙部完 ~ ~ End of Section B ~

請以最簡形式填寫答案。若計算結果是分數，請化至最簡，並確保為真分數或帶分數，或將計算結果寫成小數。 答案可以根式表示，唯該根式必須是最簡形式。除特別註明外，毋需填寫單位。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. You may use square root to represent the answer which is in the simplest form. Unless otherwise stated, no need to write down any unit. 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. 丙部：每題 7 分 Section C – each question carries 7 marks 17) 在 1 至 2019 這 2019 個自然數中最少任意挑出多少個，才必定有兩個數之差是 103？ Among the 2019 natural numbers from 1 to 2019, at least how many numbers need to be picked such that there must be two numbers having a difference of 103? 18) 宿舍分配房間，如果每個房間住 6 位學生，則多出了 12 位學生沒有房間；如果每個房間住 8 位學 生，則剛好有 5 個房間沒有人住。那麼該宿舍有多少位學生？ A dormitory is allocating the rooms. If there are 6 students living in each room, there will be 12 students not having rooms; if there are 8 students living in each room, there will be exactly 5 vacant rooms. How many students are there? 19) 圖 19 中有多少個長方形？ How many rectangles are there in Figure 19?

圖 19 Figure 19 20) 有 5 個數字，其中有兩個是相同的。當以不同的方法取出其中 4 個，這 4 個數的和可以是 163、164、 165 及 166，請問相同的是哪一個數？ There are 5 numbers, in which two of them are equal. After summing up 4 of these numbers in different ways, 4 sums are found: 163, 164, 165 and 166. Which one of these five numbers appeared twice?

~ 全卷完 ~ ~ End of Paper ~

(C) 2019 Asia International Mathematical Olympiad Union (AIMO) http://aimo.world/ All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic mechanical, photocopying, recording or otherwise, without the prior permission of Asia International Mathematical Olympiad Union.