Tài liệu Olympiad Maths Trainer 5

Terry Chew B. Sc 2 years o 1 ld 11 5 THẾ GIỚI PUBLISHERS OLYMPIAD MATHS TRAINER - 5 (11-12 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 - 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 Điện thoại: (04) 8582 5555 Hotline: 097 991 9926 Website: http://lantabra.vn http://hocgioitoan.com.vn Olympiad Maths TraineR 5 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 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 Olympiad Maths TraineR 5 CONTENTS Week 1 to Week 9  The Four Operations  Solve by Assuming  Average Problems  Catching Up  Solve by Replacement and Comparison Week 10 to Week 18  The Mathematics of Time  Area of a Composite Figure  Encountering  Divisibility  Prime Numbers Week 19 to Week 24  LCM and GCD  Venn Diagrams  The Pigeonhole Principle  Journey of the Train Week 25 Test 1 Week 26 to Week 34  Square Numbers  Value of Ones Digit  Remainder Problems  Bases other than 10  Other Operations  Algebra Week 35 to Week 43  Permutation  Combination  Number Patterns  Logic  Fractions  Comparison of Fractions Week 44 to Week 49  Ratio  Unusual Rate Problems  Percentage  Maximum and Minimum Problems Week 50 Test 2 Worked Solutions (Week 1 - Week 50) WEEK 1 Olympiad Maths Trainer 5 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 5 × 64 × 0.25 × 12.5 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? (b) 25 × 4.8 + 152 × 2.5 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. Terry Chew WEEK 1 page 1 1 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. Car A Car B Town A Town B Town C 160 km 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? Olympiad Maths Trainer - 5 WEEK 1 page 2 2 WEEK 2 Olympiad Maths Trainer 5 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 25 × 4.6 + 25 × 1.3 + 25 × 2.1 (b) 125 × 4.5 + 125 × 3.3 + 125 × 0.2 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? Terry Chew WEEK 2 page 1 3 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. 8 cm A B D C 10 cm 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? Olympiad Maths Trainer - 5 WEEK 2 page 2 4 Olympiad Maths Trainer 5 WEEK 3 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 333 333 × 333 333 (b) 333 333 × 666 666 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? Terry Chew WEEK 3 page 1 5 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? Olympiad Maths Trainer - 5 WEEK 3 page 2 6 Olympiad Maths Trainer 5 WEEK 4 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 3333 × 3334 + 2222 × 9999 (b) 99 999 × 11 111 + 33 333 × 66 667 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? Terry Chew WEEK 4 page 1 7 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. His wife bought 8 such buns and 4 such sausages for $19.60 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? Olympiad Maths Trainer - 5 WEEK 4 page 2 8 Olympiad Maths Trainer 5 WEEK 5 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 365 × 3.6 + 36.5 × 14 + 36.5 × 50 (b) 75 × 4.5 + 25 × 26.5 2. An adult movie ticket cost $7. A child movie ticket cost $5. A tourist paid $228 in all for 40 movie tickets. How many adult movie tickets were there if he had paid $32 more for the adult movie tickets? 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? Terry Chew WEEK 5 page 1 9 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? Olympiad Maths Trainer - 5 WEEK 5 page 2 10 Olympiad Maths Trainer 5 WEEK 6 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 198 × 2.4 + 19.8 × 22 + 1.98 × 540 (b) 15.8 × 68 – 15.8 × 32 – 5.8 × 36 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 $9 for her share. How much should Daniel and Eunice each get? Terry Chew WEEK 6 page 1 11 4. Peter and Randy took 16 minutes and 20 minutes respectively to cover a complete round of a circular track. They started running from two ends of the diameter of the circular track as shown in the figure below. How long did Peter take to catch up with Randy for the first time? Randy Peter 5. One particular day last year, 1 US dollar (USD) could be exchanged for 1.5 Singapore dollars (S$). 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? Olympiad Maths Trainer - 5 WEEK 6 page 2 12 Olympiad Maths Trainer 5 WEEK 7 Name: Date: Class: Marks: /24 Solve these questions. Show your working clearly. Each question carries 4 marks. 1. Compute each of the following. (a) 22 222 × 99 999 + 99 999 × 77 777 (b) 999 999 × 888 888 ÷ 666 666 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 37 What is the average weight of the four children? Terry Chew WEEK 7 page 1 13 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 Eddie spends $60 every week, Eddie will have $70 when Dion 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? Olympiad Maths Trainer - 5 WEEK 7 page 2 14