Terry Chew B. Sc
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4 THẾ GIỚI PUBLISHERS
OLYMPIAD OL YMPIAD MATHS TRAINER - 4 (10-11 years old) ALL RIGHTS RESERVED
Vietnam Vie tnam edition copyright © Sivina Education Joint stock Company 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 - 2314 - 0 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 Singapore Asia Publishers Pte Ltd và Công ty Cổ phần Giáo dục Sivina 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
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Olympiad Maths TraineR 4
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 problemsolving 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 4
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 4
CONTENTS
Week 1 to Week 9 The Four Operations Looking for a Pattern Sequence with a Common Difference Other Operations Using Models for Sum or Difference Catching up Week 10 to Week 18 The Principle of Addition The Principle of Multiplication Solve By Assuming Excess and Shortage Problems Counting Using Models for Multiplication Week 19 to Week 24 Permutation Combination Problems from Planting Trees Journey of the Train Week 25
Test 1
Week 26 to Week 34 Encountering Age Problems Solve By Replacement and Comparison Problem from Page Number Working Backwards Remainder Problems Week 35 to Week 43 Logic Number Games Solve Using Tables or Drawings Perimeter of Square and Rectangle Observation and Induction Venn Diagram Week 44 to Week 49 Average Geometry Maximum and Minimum Pigeonhole Principle Week 50
Test 2
Worked Solutions (Week 1 - Week 50)
WEEK 1
Olympiad Maths Trainer 4
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
3.
Use a simple method to calculate each of the following.
(a)
376 + 285 + 124 + 715
(b)
81 + 79 + 82 + 83 + 88
(c)
173 + 123 + 877 + 327
(d)
169 + 171 + 173 + 172 + 167
Complete each number pattern.
(a)
2, 4, 8, 16, (
), (
), (
), ···
(b)
1, 4, 9, 16, (
), (
), (
), ···
(c)
1, 1, 2, 3, 5, (
(d)
2, 1, 4, 3, 6, 5, (
), ( ), (
), (
), ···
), ···
Determine whether each of the following sequences has a common difference. Find the common difference if there is. State the rst and last terms of each sequence.
(a)
5, 10, 15, 20, 25, 30, 35
(b)
1, 4, 7, 10, 13, 16, 19
Terry Chew
WEEK 1
page 1
1
4.
(c)
1, 4, 9, 16, 25, 36, 49
(d)
2, 4, 6, 8, 10, 12, 14
If a b = (a + b) ÷ 2, evaluate
(a)
3 7,
(b)
4 (8 4).
5.
There are a total of 660 passengers on two ships. 30 passengers alight from Ship A and 70 passengers board Ship B. As a result, the number of passengers on the two ships becomes the same. How many passengers are there on each ship at rst?
6.
A car and a bicycle depart from Town A and Town B respectively at the same time.
Town A
Town B
The bicycle moves at 35 km/h and the car moves at 75 km/h. How far is Town B from Town A if the car catches up with the bicycle three hours later?
Olympiad Maths Trainer - 4
WEEK 1
page 2
2
Olympiad Maths Trainer 4
WEEK 2
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use a simple method to calculate each of the following. (a)
395 + 8 + 197 + 5 + 298 + 397
(b)
9998 + 3 + 99 + 997 + 4 + 9
(c)
9898 + 302 + 779 + 331
(d)
5678 + 543 + 123 + 477
Find the missing numbers.
3
15 27
3.
5
45 37
57
Evaluate 1 + 3 + 5 + ··· + 49.
Terry Chew
WEEK 2
page 1
3
4.
If a
b = 6 × a – 3 × b, evaluate
(a)
4
(b)
(5
5,
3)
20.
5.
Aloysius and Benjamin have $150 altogether. Aloysius’ mother gives him another $40 and Benjamin spends $10 on a book. Benjamin then has $10 more than Aloysius. How much money does each of them have at rst?
6.
A hound spotted its prey at a distance of 50 m away. It started to run towards its prey at a speed of 12 m/s but its prey could only run at a speed of 7 m/s. How long did the hound take to catch up with its prey?
Olympiad Maths Trainer - 4
WEEK 2
page 2
4
Olympiad Maths Trainer 4
WEEK 3
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use a simple method to calculate each of the following.
(a)
563 – 328 + 98 + 528
(b)
725 – 213 – 312 + 465
(c)
854 – (512 + 154) + 612
(d)
785 – (285 – 634) – 234
Find the missing numbers.
4
3.
9
7
30
29
6
5
8 3
19
Compute 2 + 4 + 6 + 8 + ··· + 98 + 100.
Terry Chew
WEEK 3
page 1
5
4.
If 3 4 = 3 + 4 + 5 + 6 = 18 and 7 3 = 7 + 8 + 9 = 24,
(a)
evaluate 2004 4.
(b)
Find the value of n when 95 n = 686.
5.
There are 120 books altogether on a bookshelf. The top shelf holds 11 more books than the middle one. The bottom shelf holds 5 books fewer than the middle one. How many books are there on each shelf?
6.
The side of a square building is 10 m long. A cat at Point A begins to chase a rat spotted at Point B and they run around the building. The cat runs at a speed of 2 m/s and the rat runs at a speed of 1 m/s. How soon will the cat catch up with the rat?
A
10 m
B
Olympiad Maths Trainer - 4
WEEK 3
page 2
6
WEEK 4
Olympiad Maths Trainer 4
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use a simple method to calculate each of the following.
(a)
77 × 6 ÷ 11
(b)
96 × 9 ÷ 12
(c)
65 × 7 ÷ 13
(d)
120 × 8 ÷ 15
Find the missing number in each of the following.
(a) 12
8
24
20
16
26
28
8
16
12
34
9
18
6
(b)
Terry Chew
WEEK 4
page 1
7
3.
Compute 1 + 2 + 3 + ··· + 99 + 100.
4.
If a
b = the remainder when a ÷ b, evaluate
(a)
2008
2007,
(b)
2010
(2000
300).
5.
If a student is transferred from Class 3A to Class 3B, the two classes will have the same number of students. If a student is transferred from Class 3B to Class 3C, Class 3C will have two more students than Class 3B. Between Class 3A and Class 3C, which class has more students?
6.
A sh swims past a kingsher at a speed of 1 m/s. The sh is 4 m away from the kingsher when the kingsher gives chase and catches it in 2 seconds. At what speed is the kingsher gliding?
Olympiad Maths Trainer - 4
WEEK 4
page 2
8
WEEK 5
Olympiad Maths Trainer 4
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use a simple method to calculate each of the following.
(a)
9 + 99 + 999 + 9999 + 99 999
(b)
28 + 298 + 2998 + 29 998 + 299 998
Observe the number pattern below: 3 __ 3 __ __ __ __ 1 , __ 1 , __ 2 , __ 1 , __ 2 , __ , 1, 2, , 4, 1
2
2
3
3
3
4
4
4
4
···
What fraction is the 35th term?
3.
Compute 3 + 7 + 11 + ··· + 95 + 99.
Terry Chew
WEEK 5
page 1
9
4.
If a U b = a × b – a + b, evaluate
(a)
2 U 4,
(b)
3 U 5.
5.
Catherine and Molly have $320 altogether. Molly and Tom have $360 in all. Tom and Catherine have $240 altogether. How much does each of them have?
6.
Mark and Nigel were jogging along a circular track. They started their jog from the same place and at the same time. Mark jogged at a speed of 220 m/min and Nigel jogged at a speed of 180 m/ min. What was the circumference of the track if Mark caught up with Nigel in 30 minutes? Mark Nigel
Olympiad Maths Trainer - 4
WEEK 5
page 2
10
Olympiad Maths Trainer 4
WEEK 6
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use distributive law to calculate each of the following.
(a)
42 × 34 + 58 × 34
(b)
37 × 54 + 63 × 54
(c)
156 × 32 – 56 × 32
(d)
233 × 46 – 133 × 46
The rst three terms of a three-number pattern are (1, 3, 6), (2, 6, 9) and (3, 9, 12). Find the sum of the three numbers in the 100th term.
3.
Find the sum of all multiples of 5 from 5 to 200.
Terry Chew
WEEK 6
page 1
11
4.
5.
If a b = a × b – (a + b), evaluate
(a)
5 6,
(b)
8 (3 4).
Some PE teachers bought a total of 83 balls for the school. The number of basketballs is twice the number of footballs. The number of volleyballs is 5 less than the number of footballs. How many balls of each type did the teachers buy?
6. start
end
1200 m
A train, which is 200 m long, travels at a speed of 20 m/s. How long does it take to pass a bridge that is 1200 m long?
Olympiad Maths Trainer - 4
WEEK 6
page 2
12
Olympiad Maths Trainer 4
WEEK 7
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
2.
Use a simple method to calculate each of the following.
(a)
999 × 222 + 333 × 334
(b)
999 × 778 + 333 × 666
Observe the number pattern below: 16, 23, 28, 38, 49, ··· What is the 6th term?
3.
Find the sum of all multiples of 7 between 100 and 200.
Terry Chew
WEEK 7
page 1
13
4.
If a b = a + (a + 1) + (a + 2) + ··· + (a + b), evaluate
(a)
3 8,
(b)
8 3.
5.
A family has four members. The father is 2 years older than the mother. The sister is 2 years older than the brother. The sum of all their present ages is 64. Three years ago, the sum of their ages was 53. How old is each of them now?
6.
Betty and Celine were jogging along a circular track surrounding a lake. The track measured 640 m. If they started from the same place and jogged in the same direction, Betty would take 16 minutes to catch up with Celine. If they jogged in the opposite direction, they would meet every 4 minutes. How long did Betty take to jog one round of the track?
Olympiad Maths Trainer - 4
WEEK 7
page 2
14
Olympiad Maths Trainer 4
WEEK 8
Name:
Date:
Class:
Marks:
/24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
Use a simple method to calculate 99 999 × 12 345.
2.
Observe the number pattern below: 121, 12 321, 1 234 321, 123 454 321, ··· What number is the 5th term?
3.
Compute 1 + 2 + 3 + ··· + 49 + 50 + 49 + ···+ 3 + 2 + 1.
Terry Chew
WEEK 8
page 1
15
4.
5.
If m n = m + (m – 1) + (m – 2) + ··· + (m – n), evaluate
(a)
7 5,
(b)
m when m 4 = 20.
In the gure below, the big square is made up of a small square and four identical rectangles. The area of the big square is 196 cm2. The area of the small square is 36 cm 2. What is the width of each rectangle?
36 cm2
6.
A train takes 27 s to cross a bridge 420 m long. It takes 30 s to pass through a tunnel 480 m long at the same speed.
(a)
What is the speed of the train?
(b)
What is the length of the train?
Olympiad Maths Trainer - 4
WEEK 8
page 2
16