a interesting book on how to solve olympiad combinatorics problems
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Olympiad
Maths
Trainer
6
CONTENTS Week 1 to Week 10 Think Algebra Venn Diagram Whole Numbers Fractions Permutation and Combination Speed I: Catching Up
Week 11 to Week 20 Speed II: Encountering Solve By Comparison and Replacement Divisibility Solve Using Table or Drawing Cryptarithm Observation and Induction
Week 21
Test 1
Week 22 to Week 31 Age Problems Percentage Ratio Area and Perimeter of Circles Rate Mathematics of Time
Week 32 to Week 41 Comparison of Fractions Pigeonhole Principle Number Pattern Logic Number Theory Maxima
Week 42
Test 2
Worked Solutions (Week 1 - Week 42)
Olympiad Maths Trainer 6
Name:
Date:
Class:
Marks:
WEEK 1 /24
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
Solve for the values of x and y in each of the following, given that x and y are whole numbers. (a)
y = 3x 8x – 2y = 8
(b)
4x + 5y = 23 9x – 5y = 3
(c)
y = 5x 3x + 2y = 65
(d)
2x + 5y = 23 6x – 5y = 9
2.
Among 64 students, 28 of them like Science, 41 like Mathematics and 20 like English. 24 of them like both Mathematics and English. 12 students like both Science and English. 10 students like both Science and Mathematics. How many students like all the three subjects?
3.
What is the value of the digit in the ones place of the following? 1 × 3 × 5 × 7 × 9 × 11 × 13 × ... × 2007 × 2009
Jonathan and Cindy run on a circular track where AB is the diameter of the track, as shown below.
A
B
If Jonathan and Cindy run towards each other at the same time from Point A and Point B respectively, it will take them 40 seconds before they meet. If they start running at the same time but in the same direction, it will take Jonathan 280 seconds to catch up with Cindy. What is the ratio of their speeds?
Solve these questions. Show your working clearly. Each question carries 4 marks. 1.
Given that a and b are whole numbers, solve for a and b in each of the following. (a) 2a + 3b = 18
(b) 4a + 6b = 68
(c) 4a + 7b = 73
(d) 3a + 8b = 47
2.
Between 1 and 2009, how many numbers are multiples of 5 or 7?
3.
The sum of the digits of a 3-digit number is 18. The tens digit is 1 more than the ones digit. If the hundreds digit and the ones digits are swapped, the difference between the new number and the original number is 396. What is the original number?
(a) Choose any three letters from a, b, c, and d. In how many ways can we arrange the three letters? (b) A teacher wants to choose a captain and vice-captain among 12 volleyball players. In how many ways can he do so?
6.
A car travelled to Town B from Town A at a constant speed of 72 km/h. It then returned from Town B to Town A at a constant speed of 48 km/h. What was the average speed of the car for the whole journey?
How many 3-digit numbers have the sum of the three digits equals to 4?
6.
A car will travel from Town A to Town B. If it travels at a constant speed of 60 km/h, it will arrive at 3.00 pm. If it travels at a constant speed of 80 km/h, it will arrive at 1.00 pm. At what speed should it be travelling if the driver aims to arrive at Town B at 2.00 pm?
