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PLEMATHEMATICALOLYMPIADTRAINING BOOKMLEVEL 6SA(12–13 years)Name:Class:Terry ChewMaster trainer at Terry Chew Academyand Co-Founder/Academic Director of
Mathematical Olympiad Training Book Level 6First Edition 2019 Singapore Asia Publishers Pte Ltd and Terry ChewPublished and Distributed by:Singapore Asia Publishers Pte LtdPLE219 Henderson Road #10-04Henderson Industrial ParkSingapore 159556Tel : 65 6276 8280Fax : 65 6276 8292Email: info.sg@sapgrp.comWebsite: www.sapgrp.comFacebook: Singapore-Asia-PublishersALL RIGHTS RESERVEDAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission ofthe SBN-10Printedin SingaporePrinted underlicence in Australia by Five Senses Education Pty LtdSAP Global Partners’ rNamibiaNepalNew ZealandSAAntigua & gyptFijiGhanaNigeriaPakistanPapua New t LuciaSaint Vincent & the GrenadinesSaudi ArabiaSeychellesSingaporeSolomon IslandsSouth AfricaSouth KoreaFor international business enquiries, email ibg@sapgrp.comwww.sapgrp.comSri LankaSyriaTanzaniaThailandTrinidad & TobagoUnited Arab EmiratesUnited KingdomUnited States of AmericaVietnamZambiaZimbabwe
1. Seven identical dominoes of size 1 cm by2 cm and with identical faces on both sidesare arranged to cover a rectangle of size2 cm 7 cm. One possible arrangementis shown below. Find the total numberof ways in which the rectangle can becovered by the seven dominoes.6. Jonathan and Cindy run on a circular trackwhere AB is the diameter of the track, asshown below.ABPLEIf Jonathan and Cindy run towards eachother at the same time from Point A andPoint B respectively, it will take them40 seconds before they meet. If theystart running at the same time but inthe same direction, it will take Jonathan280 seconds to catch up with Cindy. Whatis the ratio of their speeds?2. Among 64 students, 28 of them like Science,41 like Mathematics and 20 like English.24 of them like both Mathematics andEnglish. 12 students like both Science andEnglish. 10 students like both Science andMathematics. How many students like allthe three subjects?7. Find the value of12 – 22 32 – 42 20172 – 20182.M3. What is the value of the digit in the onesplace of the following?8. Between 1 and 2018, how many numbersare multiples of 5 or 7?SA1 3 5 7 9 11 13 . 2017 20199. The sum of the digits of a 3-digit numberis 18. The tens digit is 1 more than theones digit. If the hundreds digit and theones digits are swapped, the differencebetween the new number and the originalnumber is 396.What is the original number?4. Evaluate4 5 6 7 6 5 4 3 2 1 . 1 2 3 7777777 77777775. There are 5 dots on line A and 8 dots online B.ABFind the total number of triangles thatcan be formed using any 3 dots as theirvertices.Mathematical Olympiad Training Book Level 6 Singapore Asia Publishers Pte Ltd and Terry Chew
() () ()1 1 1 1 15. Find the value of the following.10. Evaluate 1 1 5573320172018 20182017 – 20172017 11111– 1 using a5573320182018simple method.()3 49 1 2 39 1 1 4 .16. Evaluate 29 4 52 33 4PLE11. (a) C hoose any three letters from a, b,c, and d. In how many ways can wearrange the three letters?(b) A teacher wants to choose a captain andvice-captain among 12 volleyball players.In how many ways can he do so?17. How many 3-digit numbers have the sumof the three digits equalling to 4?18. A car will travel from Town A to TownB. If it travels at a constant speed of 60km/h, then it will arrive at 3.00 pm. If ittravels at a constant speed of 80 km/h,then it will arrive at 1.00 pm. At whatspeed should it be travelling if the driveraims to arrive at Town B at 2.00 pm?M12. A car travelled to Town B from Town Aat a constant speed of 72 km/h. It thenreturned from Town B to Town A at aconstant speed of 48 km/h. What was theaverage speed of the car for the wholejourney?SA13. Lucas multiplies his month of birth by 31.He then multiplies his day of birth by 12.The sum of the two products is 213. Whenis his birthday?19. A big box can hold 48 marbles. A small boxcan hold 30 marbles. Find the number ofbig boxes and the number of small boxesthat can hold a total of 372 marbles.14. In the figure below, E and F are midpointsof AD and DC respectively. ABCD is arectangle. Find the area of the shadedregion.A20. In the figure below, the area of three circles,A, B and C, are 40 cm2, 50 cm2 and 60 cm2respectively. Given that a d 12 cm2,b d 14 cm2, c d 16 cm2 andd 8 cm2, find the area of the wholefigure.B75 cm2A25 cm2 HE55 cm2Ga80 cm2JB20 cm2DFMathematical Olympiad Training Book Level 6 Singapore Asia Publishers Pte Ltd and Terry ChewC dbcC
21. The sum of two numbers is 88. The productof the two numbers is 1612. What are thetwo numbers?26. A survey was conducted on 250 studentsregarding their preferred school activities:badminton, volleyball and basketball. 140of them liked badminton, 120 of them likedvolleyball and 100 of them liked basketball.40 of them liked both badminton and22. Evaluate volleyball, but not basketball. 20 of them1111111liked badminton and basketball, but not1 5442323volleyball. How many liked both volleyball1111111and basketball, but not badminton, given– 1 544232310 liked all three activities?using a simple method.) (() ())PLE(27. A contractor has 1088 square tiles. Inhow many ways can he form a rectangleusing all the tiles each time?23. How many ways are there to reach Bfrom A? Only movements , and areallowed.B28. Evaluate1 2 3 2 4 6 3 6 9 . 100 200 300 1 3 5 2 6 10 3 9 15 . 100 300 500Mby factorising.29. In the figure below, how many trianglescan be formed using any three points asthe vertices?25. Two fruit baskets contain some oranges. Ifan orange is transferred from the first basketto the second basket, then both basketswill have the same number of oranges. Ifan orange is transferred from the secondbasket to the first basket, then the numberof oranges in the first basket becomes thricethe number of oranges in the second basket.How many oranges are in each basket atfirst?30. A fighter plane had enough fuel to last a6-hour flight. The speed of wind and thespeed of the plane made up a total of 1500km/h when the plane was flying in thedirection of the wind during its mission.On its return trip, the total speed wasreduced to 1200 km/h as the plane wastravelling against the wind. How far couldthe plane travel before it made its return?ASA24. A tourist was travelling to a town that was60 km away. He walked at a speed of 6km/h at first. Then, he hitched a ride on ascooter travelling at 18 km/h. He arrivedat the town 4 hours from the time he setoff. How long had he walked?Mathematical Olympiad Training Book Level 6 Singapore Asia Publishers Pte Ltd and Terry Chew
31. Julie asked her teacher, “How old wereyou in 2008?” “My age in 2008 was thesum of all the digits of my year of birth,”replied the teacher. How old was theteacher in 2008?36. During a school walkathon, Alan completedthe first half of the journey at a speed of4.5 km/h. He then finished the second halfof the journey at a speed of 5.5 km/h. Onthe other hand, Benny walked at a speedof 4.5 km/h for the first half of the timetaken. He then completed the remainingjourney at 5.5 km/h. Who would arrive atthe finishing line first?(PLE32. In the figure below, the side of the squareis 14 cm. The radii of the two quadrantsare 7 cm and 14 cm respectively. A andB represent the areas of the two shaded22 . regions. Find (A – B). Take π 7)14 cmB38. In the figure below, AB 20 cm, AD 10cm and the area of quadrilateral EFGH is15 cm2.Find the total area of the shaded regions.MA37. Don and Andy have some marbles. If Dongives some marbles to Andy, then thenumber of marbles that Don has is twicewhat Andy has. If Andy gives the samenumber of marbles to Don, the numberof marbles that Don has is 4 times whatAndy has. How many marbles does eachof them have at first?33. For 12 22 32 . n2, we cancompute using n(n 1)(2n 1) 6. Findthe value of 12 22 32 . 152.ABGSA1234567834. Evaluate .123456782 – 12345677 12345679HFD35. Let m 2n – n2, where n 1, 2, 3, .,2019. How many values of m are there sothat m is divisible by 6?EC39. Alice, Bernard and Colin draw 3 cards eachfrom nine cards numbered from 1 to 9.Alice: The product of my numbers is 48.Bernard: The sum of my numbers is 15.Colin: The product of my numbers is 63.Find the three cards that each of themdraws.Mathematical Olympiad Training Book Level 6 Singapore Asia Publishers Pte Ltd and Terry Chew
To catch up:(J – C) 280 half a circumferenceEquating the two statements, we have(J C) 40 (J – C) 280J C (J – C) 7J C 7J – 7C8C 6JJ:C 8:6 4:3 The ratio of Jonathan’s speed to Cindy’s speed is4 : 3.1. I t suffices to consider four scenarios: 1 vertical, 3vertical, 5 vertical and 7 vertical.1 vertical: example4 ways3 vertical: 10 ways5 vertical: 6 ways7 vertical: 1 wayAns: 21 waysMaths24?English 12107. Use the identity a2 – b2 (a – b)(a b) (1 – 2)(1 2) (3 – 4)(3 4) (2017 – 2018)(2017 2018) –(1 2 3 4 2017 2018)(1 2018) 2018 – 2 –2 037 171SciencePLE2.28 41 20 8924 12 10 46 students are counted twice.89 – 46 43 students like either 1 or 2 subjects.Number of students 6464 – 43 21 students like all the three subjects.3. I t is sufficient to look only at the product of 1, 3,5, 7 and 9. 1 3 5 7 9 945, where the digit in theones place is 5.Hence, we have 5 5 . 5.The value of the digit in the ones place is 5.8. 5 7 352018 5 403 R 32018 7 288 R 22018 35 57 R 23Number of multiples of 5 or 7 403 288 – 57 6344. T here are 7 pairs of 7 in 1 2 . 7 6 . 1.Therefore,9. Method 1:List down all the possible numbers in the table below. 7 6 . 1 1 2 .M7777777 77777777 7 7777777 77777771 1111111 11111111 1234567654321New re, the original number is 387.Method 2:Let the 3-digit number be abc.Hence, we have cba – abc 396.100c 10b a – 100a – 10b – c 39699c – 99a 396c – a 396 99 4c 4 a----- (1)b – c 1c b – 1----- (2)Substitute (2) into (1):b–a 5b 5 a----- (3)a b c 18----- (4)Substitute (1) and (3) into (4):a 5 a 4 a 183a 9 183a 9a 9 3 3Substitute a 3 into (3): b 8Substitute a 3 into (1): c 7The original number is 387.SA5. Scenario 1: Using line A as base5 4 5C2 1 2 10Total number of triangles, 10 8C1 80.Scenario 2: Using line B as base8 7 58C2 5C1 1 2 28 5 140Ans: 220 ways6. B oth of them will cover half of a circumferenceif they run towards each other. Jonathan needsto cover half a circumference in order to catch upwith Cindy if they run in the same direction. Let the speed of Jonathon and Cindy be J and Crespectively.We can write their speeds as follows:To meet up:(J C) 40 half a circumferenceMathematical Olympiad Training Book Level 6 Singapore Asia Publishers Pte Ltd and Terry ChewOriginal numberS
TRAINING BOOK LEVEL 6 (12 13 years) Terry Chew Master trainer at Terry Chew Academy and Co-Founder/Academic Director of Name: Class: Title Page_Mathematical Olympiad Training Book
