Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 348 · 40 questions

How to solve it (Name the topic, then solve): Name the topic first: angles, counting, worst case, probability, digits, mensuration, data, coordinates, AP, algebra, primes or Venn diagrams. Use that lesson’s method; draw a quick sketch, list or Venn diagram on rough paper. Rule out options of the wrong kind — a probability above 1, or a triangle angle of 180° or more. Check by working backwards or with a second method when you can.

1. 1.A book has 165 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 330 (b) 378 (c) 387 (d) 388
2. 2.What is the sum of the first 12 terms of the AP 6, 8, 10, …? (a) 204 (b) 28 (c) 408 (d) 216
3. 3.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
4. 4.What is the smallest prime factor of 377? (a) 1 (b) 2 (c) 29 (d) 13
5. 5.What is the midpoint of the line joining (9, 0) and (13, 2)? (a) (1, 11) (b) (2, 1) (c) (22, 2) (d) (11, 1)
6. 6.A book has 76 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 134 (b) 152 (c) 144 (d) 143
7. 7.A book has 184 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 368 (b) 444 (c) 445 (d) 435
8. 8.In a class of 20 children, 10 like dance, 12 like music and 5 like both. How many like neither? (a) 8 (b) 4 (c) 5 (d) 3
9. 9.In a league, 10 teams each play every other team once. How many matches are played? (a) 45 (b) 55 (c) 100 (d) 90
10. 10.In how many different ways can the letters of the word KITE be arranged? (a) 12 (b) 24 (c) 48 (d) 16
11. 11.Which term of the AP 6, 15, 24, 33, … is 267? (a) 32nd (b) 31st (c) 29th (d) 30th
12. 12.Two angles of a triangle are 30° and 60°. What is the exterior angle at the third corner? (a) 270° (b) 150° (c) 120° (d) 90°
13. 13.There are 5 roads from town A to town B and 3 roads from town B to town C. How many different routes go from A to C through B? (a) 8 (b) 30 (c) 20 (d) 15
14. 14.What is the prime factorisation of 24? (a) 2⁴ × 3 (b) 2² × 3 (c) 2³ × 3 (d) 2 × 3³
15. 15.In an isosceles triangle, the angle between the two equal sides is 110°. What is each of the other two angles? (a) 45° (b) 35° (c) 110° (d) 70°
16. 16.Two coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/2 (b) 1/3 (c) 2/3 (d) 1/4
17. 17.What is the mean of 18, 17, 10, 14, 12 and 19? (a) 14 (b) 16 (c) 18 (d) 15
18. 18.8 T-shirts and 7 pairs of shorts are in a cupboard. How many different outfits of one T-shirt and one pair of shorts can be made? (a) 112 (b) 15 (c) 64 (d) 56
19. 19.How many two-digit numbers have digits that add up to 4? (a) 3 (b) 10 (c) 5 (d) 4
20. 20.In an isosceles triangle, the angle between the two equal sides is 140°. What is each of the other two angles? (a) 40° (b) 20° (c) 30° (d) 140°
21. 21.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
22. 22.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 160 (c) 64 (d) 256
23. 23.A bag has 4 red balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 13 (b) 16 (c) 2 (d) 5
24. 24.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 73 (d) 169
25. 25.A book has 48 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 96 (b) 87 (c) 78 (d) 88
26. 26.The numbers 5 to 13 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 30 (b) 27 (c) 81 (d) 9
27. 27.How many times is the digit 3 written when you write all the numbers from 1 to 70? (a) 17 (b) 16 (c) 7 (d) 18
28. 28.Two angles of a triangle are 65° and 50°. What is the exterior angle at the third corner? (a) 245° (b) 115° (c) 130° (d) 65°
29. 29.A cuboid is 4 cm long, 2 cm wide and 3 cm high. What is its total surface area? (a) 52 sq cm (b) 26 sq cm (c) 104 sq cm (d) 24 sq cm
30. 30.A fair dice is rolled once. What is the probability of getting a factor of 6? (a) 1/3 (b) 5/6 (c) 1/4 (d) 2/3
31. 31.In a class of 39 children, 17 like dance, 24 like music and 12 like both. How many like only dance? (a) 15 (b) 17 (c) 5 (d) 12
32. 32.A fair dice is rolled once. What is the probability of getting a number greater than 1? (a) 1/3 (b) 1/6 (c) 1/5 (d) 5/6
33. 33.How many times is the digit 8 written when you write all the numbers from 1 to 79? (a) 6 (b) 9 (c) 7 (d) 8
34. 34.What is the smallest number of children in a group that makes sure at least two of them were born in the same day of the week? (a) 8 (b) 15 (c) 7 (d) 14
35. 35.What is the mode of 1, 18, 18, 6, 6, 18 and 5? (a) 5 (b) 6 (c) 1 (d) 18
36. 36.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
37. 37.If every edge of a cube is made 2 times as long, how many times as big does its surface area become? (a) 2 times (b) 8 times (c) 5 times (d) 4 times
38. 38.What is the circumference of a circle of radius 35 cm? (Use π = 22/7.) (a) 440 cm (b) 220 cm (c) 110 cm (d) 3850 cm
39. 39.Two dice are rolled together. What is the probability that the two numbers add up to 6? (a) 5/11 (b) 5/36 (c) 1/6 (d) 1/11
40. 40.What is the smallest number of children in a group that makes sure at least four of them were born in the same month? (a) 48 (b) 37 (c) 36 (d) 49

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