Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 571 · 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 fair dice is rolled once. What is the probability of getting an even number? (a) 2/3 (b) 1/2 (c) 1/3 (d) 1/6
2. 2.A drawer has lots of blue and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 2 (c) 4 (d) 3
3. 3.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
4. 4.Two angles of a triangle are 45° and 80°. What is the exterior angle at the third corner? (a) 135° (b) 125° (c) 55° (d) 100°
5. 5.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 20 (d) 10
6. 6.What is the prime factorisation of 336? (a) 2³ × 3 × 7 (b) 2⁴ × 3 × 7 (c) 2⁵ × 3 × 7 (d) 2 × 3 × 7⁴
7. 7.Two angles of a triangle are 45° and 95°. What is the third angle? (a) 140° (b) 85° (c) 40° (d) 135°
8. 8.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
9. 9.In a class of 32 children, 18 like cricket, 19 like football and 8 like both. How many like only cricket? (a) 18 (b) 11 (c) 10 (d) 13
10. 10.A bag has 3 yellow, 4 blue and 2 red balls. One ball is taken out without looking. What is the probability that it is not red? (a) 8/9 (b) 7/9 (c) 2/9 (d) 1/3
11. 11.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 3 and 21 (c) 1 and 15 (d) 7 and 9
12. 12.How far is the point (6, 8) from the origin (0, 0)? (a) 14 (b) 11 (c) 10 (d) 100
13. 13.What is the sum of the first 5 terms of the AP 9, 12, 15, …? (a) 75 (b) 150 (c) 21 (d) 78
14. 14.6 T-shirts and 2 pairs of shorts are in a cupboard. How many different outfits of one T-shirt and one pair of shorts can be made? (a) 12 (b) 8 (c) 18 (d) 24
15. 15.A drawer has lots of red, blue, white, black and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 10 (b) 6 (c) 2 (d) 5
16. 16.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
17. 17.Two angles of a triangle are 75° and 75°. What is the exterior angle at the third corner? (a) 105° (b) 30° (c) 210° (d) 150°
18. 18.A bag has 3 green, 4 blue and 4 yellow balls. One ball is taken out without looking. What is the probability that it is green? (a) 3/8 (b) 8/11 (c) 3/11 (d) 1/3
19. 19.A drawer has lots of red, yellow, black, white and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 6 (b) 2 (c) 10 (d) 5
20. 20.What is the sum of the first 6 terms of the AP 8, 14, 20, …? (a) 156 (b) 38 (c) 138 (d) 276
21. 21.How many two-digit numbers have digits that add up to 4? (a) 3 (b) 4 (c) 5 (d) 10
22. 22.How many two-digit numbers have digits that add up to 2? (a) 3 (b) 10 (c) 1 (d) 2
23. 23.The numbers 6 to 14 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 90 (b) 33 (c) 10 (d) 30
24. 24.A bag has 3 black balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 6 (b) 4 (c) 8 (d) 2
25. 25.In a league, 5 teams each play every other team once. How many matches are played? (a) 15 (b) 10 (c) 20 (d) 25
26. 26.What is the sum of the first 11 terms of the AP 5, 9, 13, …? (a) 45 (b) 297 (c) 550 (d) 275
27. 27.What is the smallest number of children in a group that makes sure at least four of them were born in the same day of the week? (a) 21 (b) 29 (c) 28 (d) 22
28. 28.How many times is the digit 4 written when you write all the numbers from 1 to 62? (a) 6 (b) 17 (c) 15 (d) 16
29. 29.In how many different ways can the letters of the word CAT be arranged? (a) 12 (b) 6 (c) 3 (d) 9
30. 30.Two coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/4 (b) 1/3 (c) 1/2 (d) 2/3
31. 31.What is the circumference of a circle of diameter 56 cm? (Use π = 22/7.) (a) 88 cm (b) 176 cm (c) 352 cm (d) 2464 cm
32. 32.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 15 (b) 22 (c) 21 (d) 14
33. 33.What is the median of 18, 1, 20, 7, 25, 8 and 24? (a) 20 (b) 18 (c) 8 (d) 7
34. 34.How many times is the digit 6 written when you write all the numbers from 1 to 45? (a) 3 (b) 2 (c) 4 (d) 5
35. 35.How many times is the digit 9 written when you write all the numbers from 1 to 61? (a) 6 (b) 4 (c) 7 (d) 5
36. 36.The numbers 18 to 26 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 22 (b) 69 (c) 66 (d) 198
37. 37.Two angles of a triangle are 75° and 40°. What is the exterior angle at the third corner? (a) 105° (b) 115° (c) 140° (d) 65°
38. 38.What is the mode of 15, 5, 5, 6, 6, 5 and 12? (a) 12 (b) 5 (c) 15 (d) 6
39. 39.A book has 231 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 576 (b) 462 (c) 586 (d) 585
40. 40.A bag has 9 yellow balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 12 (c) 20 (d) 10

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