Olympiad Maths Trainer 5

Olympiad Maths Trainer là bộ sách bài tập Toán gồm 6 cuốn, dành cho học sinh từ 7 13 tuổi. Bộ sách được biên soạn bởi Terry Chew – thầy giáo có nhiều năm kinh nghiệm trong việc ôn luyện cho học sinh tham gia các kỳ thi Toán quốc tế và là tác giả nổi tiếng tại Singapore. Ông luôn hướng tới việc tìm kiếm giải pháp để giải quyết các bài toán khó một cách đơn giản và hiệu quả nhất. Các cuốn sách sẽ cung cấp bài tập thực hành cho các dạng toán khác nhau đã được giới thiệu trong bộ sách đầu tiên của tác giả là bộ “Đánh thức Tài năng Toán học” (Unleash the Maths Olympian in You). Olympiad Maths Trainer hữu ích đối với: – Học sinh: muốn cải thiện khả năng giải quyết các vấn đề trong quá trình học toán, hướng tới mục tiêu tham gia Olympic Toán học và các kỳ thi tương tự. – Giáo viên: những người luôn tìm kiếm nguồn học liệu để ôn luyện cho học trò tham gia các kỳ thi Toán mang tầm quốc tế. – Các bậc phụ huynh: mong muốn được đồng hành cùng con trong việc học toán và chinh phục các kỳ thi Toán.

ALL RIGHTS RESERVED

Vietnam edition copyright © Sivina Education Joint stock Company, 2016.

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 the publishers.

ISBN: 978 - 604 - 77 - 2315 - 7

Printed in Viet Nam

Bản quyền tiếng Việt thuộc về Công ty Cổ phần Giáo dục Sivina, xuất bản theo hợp đồng chuyển nhượng bản quyền giữa Singapore Asia Publishers Pte Ltd và Công ty Cổ phần Giáo dục Sivina 2016.

Bản quyền tác phẩm đã được bảo hộ, mọi hình thức xuất bản, sao chụp, phân phối dưới dạng in ấn, văn bản điện tử, đặc biệt là phát tán trên mạng internet mà không được sự cho phép của đơn vị nắm giữ bản quyền

là hành vi vi phạm bản quyền và làm tổn hại tới lợi ích của tác giả và đơn vị đang nắm giữ bản quyền Không ủng hộ những hành vi vi phạm bản quyền Chỉ mua bán bản in hợp pháp.

ĐƠN VỊ PHÁT HÀNH:

Công ty Cổ phần Giáo dục Sivina

Địa chỉ: Số 1, Ngõ 814, Đường Láng, Phường Láng Thượng, Quận Đống Đa, TP Hà Nội

I first met Terry when he approached SAP to explore the possibility of publishing Mathematical Olympiad type questions that he had researched, wrote and compiled What struck me at our first meeting was not the elaborate work that he had consolidated over the years while teaching and training

students, but his desire to make the materials accessible to all students,

including those who deem themselves “not so good” in mathematics Hence

the title of the original series was most appropriate: Maths Olympiad — Unleash the Maths Olympian in You!

My understanding of his objective led us to endless discussions on how to make the book easy to understand and useful to students of various levels It was in these discussions that Terry demonstrated his passion and creativity in solving non-routine questions He was eager to share these techniques with his students and most importantly, he had also learned alternative methods of solving the same problems from his group of bright students

This follow-up series is a result of his great enthusiasm to constantly sharpen his students’ mathematical problem-solving skills I am sure those

who have worked through the first series, Maths Olympiad — Unleash the Maths Olympian in You!, have experienced significant improvement in their

problem-solving skills Terry himself is encouraged by the positive feedback and delighted that more and more children are now able to work through non-routine questions

And we have something new to add to the growing interest in

Mathematical Olympiad type questions — Olympiad Maths Trainer is now on Facebook! You can connect with Terry via this platform and share

interesting problem-solving techniques with other students, parents and teachers

I am sure the second series will benefit not only those who are preparing for mathematical competitions, but also all who are constantly looking for additional resources to hone their problem-solving skills

Olympiad Maths TraineR 5

A word from

the author

Dear students, teachers and parents,

Welcome once more to the paradise of Mathematical Olympiad where the enthusiastic young minds are challenged by the non-routine and exciting mathematical problems!

My purpose of writing this sequel is twofold

The old adage that “to do is to understand” is very true of mathematical learning This series adopts a systematic approach to provide practice for the various types of mathematical problems introduced in my first series

of books

In the first two books of this new series, students are introduced to 5 different types of mathematical problems every 12 weeks They can then apply different thinking skills to each problem type and gradually break certain mindsets in problem-solving The remaining four books comprise 6 different types of mathematical problems in the same manner In essence, students are exposed to stimulating and interesting mathematical problems where they can work on creatively

Secondly, the depth of problems in the Mathematical Olympiad cannot be underestimated The series contains additional topics such as the Konigsberg Bridge Problem, Maximum and Minimum Problem, and

some others which are not covered in the first series, Maths Olympiad – Unleash the Maths Olympian in You!

Every student is unique, and so is his or her learning style Teachers and parents should wholly embrace the strengths and weaknesses of each student in their learning of mathematics and constantly seek improvements

I hope you will enjoy working on the mathematical problems in this series just as much as I enjoyed writing them

Terry Chew

 The Mathematics of Time

 Area of a Composite Figure

 The Pigeonhole Principle

 Journey of the Train

Week 25 Test 1

Olympiad Maths TraineR 5

CONTENTS

2 A one-way bus ticket cost $25 A two-way bus ticket cost $35 A

cashier collected $560 altogether from the sale of 20 tickets How many one-way bus tickets were sold?

3 The average of eight numbers is 29 The average of the first five

numbers is 26 The average of the last four numbers is 31 Find the fifth number

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4 Town A and Town B are 160 km apart Car A departs from Town

A and Car B departs from Town B at the same time Both cars are heading towards Town C

The speeds of Car A and Car B are 80 km/h and 60 km/h respectively How long does Car A take to catch up with Car B?

5 1 carton of apples and 2 cartons of oranges weigh 125 kg 2

cartons of pineapples and 2 cartons of oranges weigh 220 kg

3 cartons of apples and 2 cartons of pineapples weigh 235 kg How much does 1 carton of pineapples weigh?

6 The sum of a, b and c is 14 If 9a + 8b + 6c = 101

, what is the value of 2a + b – c?

2 50 vans and lorries have 228 wheels Given that each lorry has

6 wheels, how many lorries are there?

3 The scores of four students in a mathematics test are 79, 83,

92 and 80 Aureila’s score is 6 more than the average score of the four students and her score What is Aureila’s score in the mathematics test?

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4 Both Leon and Mark took part in a 100-m race When Leon was

at the 84-m mark, Mark had 30 m to go before finishing the race How far was Mark from the finishing line when Leon reached the finishing line?

5 In the figure below, the shaded region is a square Find the

perimeter of rectangle ABCD

6 By selling a laptop at its original price, James could make a profit

of $500 Since the model of the laptop was outdated, he sold

it at 80% of its original price As a result, he incurred a loss of

$400 How much did James buy the laptop?

WEEK 2

2 Adeline and Bryan took part in a mathematics quiz There were

30 questions altogether 5 marks would be given to each correct answer A penalty of 3 marks would be given to each wrong answer The total score of Adeline and Bryan was 236 How many questions did Adeline answer correctly if she had answered

2 correct questions more than Bryan?

3 In a choir, the number of girls is twice the number of boys

The average height of the boys is 150 cm and the average height of the girls is 159 cm What is the average height of all the choir members?

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4 A hound caught a glimpse of its prey that was 36 m

away It ran towards its prey immediately and was only 12 m behind its prey after it had run a distance of

48 m How many metres more would the hound need to run to catch up with its prey? (Assume that they ran at constant speeds.)

5 It took Samuel 18 minutes to walk 1200 m and run 1600 m It

took him 17 minutes to walk 600 m and run 2400 m

(a) How fast could Samuel walk?

(b) How far could Samuel run in 4 minutes?

6 The soldiers could march 25 km on a dry day and 20 km on a

rainy day Given that they marched a total of 480 km in 20 days, how many rainy days were there?

WEEK 3

2 A man ran and cycled a total of 12 600 m in 45 minutes His

running and cycling speeds were 180 m/min and 360 m/min respectively How long had he been cycling?

3 Bridget’s grandmother lived 720 m away On a Saturday morning,

Bridget walked to her grandmother’s house at a speed of 60 m/min and walked back home at a speed of 48 m/min What was her average speed for the whole journey?

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4 Julie and Kelly were cycling along a 2400-m circular track If

they started cycling from the same place but in the opposite direction, it would take them 4 minutes to meet each other

If they started cycling from the same place and in the same direction, Julie would take 24 minutes to catch up with Kelly Find their cycling speeds

5 Mr Greenwood paid $11 for 5 buns and 2 sausages last week

this week

(a) How much was each bun?

(b) How much was each sausage?

6 A car was travelling at a constant speed from Town A to Town

B If it travelled at 60 km/h, it would reach Town B 4 hours later than scheduled If it travelled at 70 km/h, it would reach Town B

2 hours later than scheduled How far was Town B from Town A?

WEEK 4

3 Douglas’ average score for four tests is 68 What should his

score be for the fifth test if he wishes to have an average score

of 72 for all five tests?

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4 Megan’s father realised that she had left her science textbook

at home He chased after Megan 5 minutes after she had left for school He returned home right after passing the textbook

to her His constant speed for the whole journey was 110 m/min How far was the school from Megan’s home if he had reached home the same time Megan had arrived at the school? Assume Megan walked at a constant speed of

55 m/min

5  +  +  =  + 

D + D =  +  + 

 +  + D +  = 575

Find the values of ,  and D

6 Mr Thomas walks to his office every morning He starts work at

8 am If he walks at a speed of 80 m/min, he will arrive at his office 10 minutes earlier If he walks at a speed of 70 m/min, he will arrive at his office 6 minutes earlier

(a) At what time must he leave his house in order to arrive at his

office just on time?

(b) How far is his office from his home?

WEEK 5

2 A farmer has some chickens and rabbits The animals make up

a total of 106 legs If all the rabbits were turned into chickens and all the chickens were turned into rabbits, there would be 116 legs How many chickens and how many rabbits are there?

3 Cecilia, Daniel and Eunice shared five hamburgers equally Three

burgers were bought by Daniel while the other two burgers were bought by Eunice Cecilia paid Daniel and Eunice a total of $9 for her share How much should Daniel and Eunice each get?

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4 Peter and Randy took 16 minutes and 20 minutes respectively to

cover a complete round of a circular track They started running from two ends of the diameter of the circular track as shown in the figure below How long did Peter take to catch up with Randy for the first time?

5 One particular day last year, 1 US dollar (USD) could be exchanged

for 1.5 Singapore dollars (S$) On the same day, one could buy

550 Hong Kong dollars (HKD) with 100 Singapore dollars How much HKD could be bought with 80 USD that day?

6 Some school buses were hired for a pre-holiday excursion If

school buses with 35 seats were hired, 10 students would not get to board the bus If buses with 40 seats were hired instead, there would be an extra bus

(a) How many buses were hired?

(b) How many students were going on the excursion?

WEEK 6 Peter

Randy

2 Andrea, Bella and Clifford took part in a quiz that consisted of 10

questions Each correct answer carried 10 marks The deduction for every wrong answer was 3 marks The scores obtained by Andrea, Bella and Clifford were 87, 61 and 9 respectively How many questions did each of them answer correctly?

3 Four children take turns to weigh themselves in pairs All the six

readings of the total weight of every two children are recorded as follows:

39 34 38 38 42 37What is the average weight of the four children?

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4 A parcel has to reach its destination by 9 am A postman is

assigned to deliver the parcel from the post office If the postman travels on his scooter at 60 km/h, he will be late by 15 minutes

If he travels at 72 km/h, he will arrive at the destination at 8.50

am How far does the postman need to travel in order to deliver the parcel?

5 An artist and his assistant were to make a certain number of

pieces of handicraft together They originally planned to finish this task in 20 days In the end, the artist ended up producing 5 more such pieces of handicraft each day His assistant, on the other hand, produced 3 more such pieces of handicraft each day

As a result, they were able to finish their work in 16 days How many pieces of handicraft were made?

6 Dion and Eddie have some savings each If Dion spends $30 and

has used up all her savings If Eddie spends $30 and Dion spends

$60 every week, Eddie will have $520 when Dion has used up all her savings How much does each of them have?

WEEK 7

2 Colin cycled to his grandmother’s house at a speed of 240 m/min

He cycled back home at a speed of 180 m/min What was his average speed for the whole journey?

3 The average score obtained by a group of students in an English

test was 92 Among the students, the boys scored an average

of 90 and the girls scored an average of 93.5 How many boys were there if there were 20 girls?

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4 Alfred cycled from his home to a park at a speed of 300 m/min

He then cycled back home at a speed of 270 m/min What was his average speed for the whole journey?

5 There was a certain number of apples at a wholesale centre

If the apples were sold at $2.50 per kg, the wholesaler would lose $50 If the apples were sold at $3 per kg, the wholesaler would earn $90 How much should 1 kg of apples be priced if the wholesaler wanted to earn $62 from the sale of all the apples?

6 Aunt Rita uses canola and olive oil for cooking If she is to use

50 g of canola oil and 25 g of olive oil every day, she will have

150 g of canola oil when the olive oil is finished If she is to use 50 g of olive oil and 25 g of canola oil every day, 420 g

of canola oil will be left when the olive oil is finished How much oil of each type does she have at first?

WEEK 8