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).
Terry Chew B Sc THẾ GIỚI PUBLISHERS d 10 years o l OLYMPIAD MATHS TRAINER - (9-10 years old) 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 - 2313 - Printed in Viet Nam Bản quyền tiếng Việt thuộc Công ty Cổ phần Giáo dục Sivina, xuất theo hợp đồng chuyển nhượng quyền Singapore Asia Publishers Pte Ltd Công ty Cổ phần Giáo dục Sivina 2016 Bản quyền tác phẩm bảo hộ, hình thức xuất bản, chụp, phân phối dạng in ấn, văn điện tử, đặc biệt phát tán mạng internet mà không cho phép đơn vị nắm giữ quyền hành vi vi phạm quyền làm tổn hại tới lợi ích tác giả đơn vị nắm giữ quyền Không ủng hộ hành vi vi phạm quyền Chỉ mua bán 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 Điện thoại: (04) 8582 5555 Hotline: 097 991 9926 Website: http://lantabra.vn http://hocgioitoan.com.vn Olympiad Maths TraineR FOREWORD 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 Michelle Yoo Chief Publisher SAP Olympiad Maths TraineR 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 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 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 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 Olympiad Maths TraineR CONTENTS Week to Week Konigsberg Bridge Problems Geometric Patterns IQ Maths Solve using the Shortest Route Logic The Story of Gauss Week 10 to Week 18 Solve Differences and Sums Solve Problems on Multiples Age Problems Working Backwards Counting Looking for a Pattern Week 19 to Week 24 Tricks in Multiplication Problems from Planting Trees Number Puzzles Page–number Problems Week 25 Test Week 26 to Week 34 Tricks in Addition and Subtraction Catching up Encountering Finding Perimeter Excess–and–Shortage Problems Pigeonhole Principle Week 35 to Week 43 Chicken–and–Rabbit Problems Solve By Replacement Make a List or Table Solve By Comparison Geometry Cryptarithm Week 44 to Week 49 Remainder Problems Time Average Problems Areas of Square and Rectangle Week 50 Test Worked Solutions (Week - Week 50) Olympiad Maths Trainer WEEK Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Are you able to trace the figure below without lifting your fingers off the paper? You are not allowed to trace any line segments more than once Draw the next pattern ? It takes minutes to fry a pancake One side of the pancake takes minutes to fry and the other side takes only minutes Two pancakes can be placed on the frying pan at a time What is the shortest time to fry all five pancakes? Terry Chew WEEK page 1 How many different ways are there for Eton to go to the library if he can only take the routes indicated by the arrows? Eton library Among William, Sarah and Hayden, only one of them watched the movie ‘Harry Potter and the Order of the Phoenix’ When Jane asked the three friends about the movie, they gave her the following answers William: Sarah watched the movie already Sarah: I haven’t got a chance to watch it Hayden: Maybe I will watch it next week Only one of them told the truth Who had watched the movie? If William had watched the movie, Lie If Sarah had watched the movie, Truth William Sarah Hayden Lie Truth William Sarah Hayden If Hayden had watched the movie, Lie Truth William Sarah Hayden Compute each of the following using a simple method (a) + + + + + (b) + + + + 10 + 12 (c) + + + + 11 + 13 (d) + + 13 + 18 + 23 + 28 Olympiad Maths Trainer WEEK page 2 Olympiad Maths Trainer WEEK Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Are you able to trace the figure below without lifting your fingers off the paper? You are not allowed to trace any line segments more than once Draw the missing pattern in the box below All the Primary students at Russels Elementary School subscribe to at least one magazine 150 students subscribe to Wildlife 208 students subscribe to A-Star Maths 88 students subscribe to both magazines How many Primary students are there at Russels Elementary School? Terry Chew WEEK page In the figure below, each letter is connected to another by a straight line How many different ways are there to form the word “FORTUNE”? A straight line must connect two letters at all times T R U O T N F R U E O T N R U T The prince has hidden the princess’ diamond ring in one of the three jewellery boxes Each box is labelled as follows: Box A: The ring is not in here Box B: This box is empty Box C: The ring is in Box A Only one jewellery box has the correct label Help the princess to find the ring If the ring is in A, Right Wrong A B C Right Wrong A B C If the ring is in C, Right If the ring is in B, Wrong A B C Compute each of the following using a simple method (a) + + + + ··· + + 10 (b) + + + + + 11 + 13 + 15 (c) + + + + 10 + 12 + 14 + 16 (d) 11 + 12 + 13 + ··· + 19 + 20 Olympiad Maths Trainer WEEK page Olympiad Maths Trainer WEEK Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Is it possible to trace the figure below without lifting your fingers off the paper? You are not allowed to trace any line segments more than once Draw the missing pattern in the box below Lina’s granny rears a hen that lays an egg every day She then cooks eggs for Lina every morning On 1st May, her granny has collected 20 eggs How long can the eggs last them? Terry Chew WEEK page How many different ways are there for the car to travel to the city? Assume the car could only travel in the direction of South or East city Alice, Beatrice and Charlene were born in Canada, Korea and Thailand, but not in that order Alice has never been to Canada Beatrice was not born in Canada, and neither was she born in Thailand Find their places of birth Canada Korea Thailand Alice Beatrice Charlene Compute each of the following using a simple method (a) + + + + ··· + 19 + 20 (b) + + + + ··· + 17 + 19 (c) + + + + ··· + 18 + 20 (d) 21 + 22 + 23 + 24 + ··· + 39 + 40 Olympiad Maths Trainer WEEK page Olympiad Maths Trainer WEEK Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Is it possible to trace the figure below without lifting your fingers off the paper? You are not allowed to trace any line segments more than once Draw the missing pattern in the box below Wilfred bought a terrier at a price of $200 He then sold it for $250 He later bought it back at a price of $280 and then sold it for $330 How much did Wilfred make in all? Terry Chew WEEK page In the figure below, each letter is connected to another by a straight line How many different ways are there to form the word “COMPUTE”? Each straight line must connect two letters P M U O P T C M U E O P T M U P Each of three boxes contains either two red, one blue and one red, or two blue balls The following shows the labels on the three boxes Box A: Two red balls Box B: Two blue balls Box C: One blue and one red balls All the boxes have been wrongly labelled George is able to rectify the situation by picking out a ball from one of the boxes Explain how George is able to that Compute each of the following using a simple method (a) + + + + ··· + 49 + 50 (b) + + + + ··· + 48 + 50 (c) + + + + ··· + 47 + 49 Olympiad Maths Trainer WEEK page WEEK Olympiad Maths Trainer Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks A rat is trapped in a small maze Help the rat to find its way, provided it must pass through each door exactly once What is the next pattern? ? Alicia took days to finish reading a book Her sister took days to finish reading the same book If Alicia were to read 15 pages more than her sister every day, what was the total pages of the book? Terry Chew WEEK page How many different ways are there for the ant to return home? Assume it could only travel towards the north and east home David, Julie and Mary are designer, writer and violinist, but not in this order Mary is older than the violinist David and the writer are not of the same age The writer is younger than Julie Find their jobs Designer Writer Violinist David Julie Mary The sum of six consecutive numbers is 123 Find the first number of this sequence Olympiad Maths Trainer WEEK page 10 Olympiad Maths Trainer WEEK Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Part of a recreation park has the following path Show how a jogger can cover the whole path exactly once Draw the missing pattern in the box below Andrew, Bryan and Charlie each draws two cards from a stack of cards numbered to One of Bryan’s cards has a number twice of the other The sum of the numbers on Charlie’s cards is The sum of the numbers on Andrew’s cards is but the difference is not Which two cards are not drawn? Terry Chew WEEK page 11 How many different ways are there for the construction worker to go to Site A if he must avoid the dangerous Site B? Assume he can go → and ↓ only B A Complete the number pattern below 16 26 42 10 178 110 Compute each of the following using a simple method (a) 100 – 99 + 98 – 97 + 96 – 95 + ··· + 50 – 49 (b) + + + + ··· + 99 + 100 (c) 200 – 196 + 192 – 188 + 184 – 180 + ··· + 128 – 124 Olympiad Maths Trainer WEEK page 12 WEEK Olympiad Maths Trainer Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks There are two small islands in the middle of a river Seven bridges are built to link the islands to the river banks as shown below river bank river island island river bank Show how a visitor can cross all the seven bridges exactly once What is the next pattern? ? page WEEK Terry Chew What number is opposite each number? 13 A spider lies in ambush for the ant as shown below How many different ways are there for the ant to reach home safely? Assume the ant can only move in the directions of → and ↑ home Many years ago, the number of Saturdays was more than that of Fridays in a particular month Similarly, the number of Sundays was more than that of Mondays On which day of the week was the eleventh day in that month? Sun Mon Tue Wed Thu Fri Sat A theatre has 15 rows of seats The first row has 10 seats The second row has more seats than the first row The third row has more seats than the second row and so on How many seats are there altogether in the theatre? Olympiad Maths Trainer WEEK page 14 WEEK Olympiad Maths Trainer Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries marks Six events were held at an exhibition hall A visitor could walk from one event hall to another by passing through the doors as shown in the figure below Show how a visitor could visit all the six events by passing through each door exactly once Hall F Hall E Hall A Hall D Hall B Hall C Shade the third pattern correctly Anne and Betty want to buy a book Anne is short of 50¢ and Betty is short of $4.50 When they pool their money, the total amount is still not enough to buy the book How much is the book? Assume 10¢ is the smallest unit Terry Chew WEEK page 15 How many different ways are there to go from A to B if only movements in the directions of → and ↑ are allowed? B A Megan, Nikita, Patsy and Stella live on the second, third, fourth and fifth floors of a six-storey apartment, but not in this order Their professions are artist, pianist, engineer and sales executive Megan lives on the floor higher than that of Nikita but lower than that of Patsy Stella lives on the fifth floor The sales executive lives one floor above the engineer but one floor lower than the pianist The artist lives on the lowest floor Find out their professions and the floor where each of them lives The sum of eight consecutive odd numbers is 192 Find the last number of the sequence Olympiad Maths Trainer WEEK page 16 [...]... Korea Thailand Alice Beatrice Charlene 6 Compute each of the following using a simple method (a) 1 + 2 + 3 + 4 + ··· + 19 + 20 (b) 1 + 3 + 5 + 7 + ··· + 17 + 19 (c) 2 + 4 + 6 + 8 + ··· + 18 + 20 (d) 21 + 22 + 23 + 24 + ··· + 39 + 40 Olympiad Maths Trainer 3 WEEK 3 page 2 6 Olympiad Maths Trainer 3 WEEK 4 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question... one of the boxes Explain how George is able to do that 6 Compute each of the following using a simple method (a) 1 + 2 + 3 + 4 + ··· + 49 + 50 (b) 2 + 4 + 6 + 8 + ··· + 48 + 50 (c) 1 + 3 + 5 + 7 + ··· + 47 + 49 Olympiad Maths Trainer 3 WEEK 4 page 2 8 WEEK 5 Olympiad Maths Trainer 3 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries 4 marks 1 A rat... numbers is 1 23 Find the first number of this sequence Olympiad Maths Trainer 3 WEEK 5 page 2 10 Olympiad Maths Trainer 3 WEEK 6 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries 4 marks 1 Part of a recreation park has the following path Show how a jogger can cover the whole path exactly once 2 Draw the missing pattern in the box below 3 Andrew, Bryan... Wed Thu Fri Sat 6 A theatre has 15 rows of seats The first row has 10 seats The second row has 3 more seats than the first row The third row has 3 more seats than the second row and so on How many seats are there altogether in the theatre? Olympiad Maths Trainer 3 WEEK 7 page 2 14 WEEK 8 Olympiad Maths Trainer 3 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question... following using a simple method (a) 100 – 99 + 98 – 97 + 96 – 95 + ··· + 50 – 49 (b) 1 + 2 + 3 + 4 + ··· + 99 + 100 (c) 200 – 196 + 192 – 188 + 184 – 180 + ··· + 128 – 124 Olympiad Maths Trainer 3 WEEK 6 page 2 12 WEEK 7 Olympiad Maths Trainer 3 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries 4 marks 1 There are two small islands in the middle of a river.. .Olympiad Maths Trainer 3 WEEK 3 Name: Date: Class: Marks: /24 Solve these questions Show your working clearly Each question carries 4 marks 1 Is it possible to trace the figure below without lifting your fingers off the paper? You are not allowed to trace any line segments more than once 2 Draw the missing pattern in the box below 3 Lina’s granny rears a hen that... banks as shown below river bank river island island river bank Show how a visitor can cross all the seven bridges exactly once 2 What is the next pattern? ? 2 page 1 3 4 WEEK 7 1 Terry Chew 3 2 6 5 4 3 What number is opposite each number? 13 4 A spider lies in ambush for the ant as shown below How many different ways are there for the ant to reach home safely? Assume the ant can only move in the directions... The artist lives on the lowest floor Find out their professions and the floor where each of them lives 6 The sum of eight consecutive odd numbers is 192 Find the last number of the sequence Olympiad Maths Trainer 3 WEEK 8 page 2 16 ... your fingers off the paper? You are not allowed to trace any line segments more than once 2 Draw the missing pattern in the box below 3 Wilfred bought a terrier at a price of $200 He then sold it for $250 He later bought it back at a price of $280 and then sold it for $33 0 How much did Wilfred make in all? Terry Chew WEEK 4 page 1 7 4 In the figure below, each letter is connected to another by a straight... not 3 Which two cards are not drawn? Terry Chew WEEK 6 page 1 11 4 How many different ways are there for the construction worker to go to Site A if he must avoid the dangerous Site B? Assume he can go → and ↓ only B A 5 Complete the number pattern below 16 26 2 42 2 10 178 6 110 4 6 Compute each of the following using a simple method (a) 100 – 99 + 98 – 97 + 96 – 95 + ··· + 50 – 49 (b) 1 + 2 + 3 +