Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 45 · 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.In a class of 40 children, 25 like maths, 19 like science and 6 like both. How many like only maths? (a) 25 (b) 21 (c) 13 (d) 19
2. 2.A drawer has lots of 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) 3 (b) 4 (c) 5 (d) 2
3. 3.What is the prime factorisation of 504? (a) 2³ × 3² × 7 (b) 2² × 3² × 7 (c) 2⁴ × 3² × 7 (d) 2 × 3² × 7³
4. 4.What is the area of a circle of radius 14 cm? (Use π = 22/7.) (a) 308 sq cm (b) 88 sq cm (c) 616 sq cm (d) 2464 sq cm
5. 5.What is the sum of the first 5 terms of the AP 1, 5, 9, …? (a) 17 (b) 90 (c) 55 (d) 45
6. 6.What is the sum of the first 7 terms of the AP 7, 8, 9, …? (a) 70 (b) 13 (c) 140 (d) 71
7. 7.There are 3 roads from town A to town B and 6 roads from town B to town C. How many different routes go from A to C through B? (a) 36 (b) 18 (c) 21 (d) 9
8. 8.Two dice are rolled together. What is the probability that the two numbers add up to 3? (a) 2/11 (b) 1/12 (c) 1/11 (d) 1/18
9. 9.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 36 (d) 6
10. 10.What is the midpoint of the line joining (10, 9) and (12, 5)? (a) (22, 14) (b) (1, 2) (c) (11, 7) (d) (7, 11)
11. 11.In a class of 39 children, 12 like tea, 19 like coffee and 2 like both. How many like at least one of the two? (a) 27 (b) 31 (c) 39 (d) 29
12. 12.A bag has 5 blue, 6 green and 8 yellow balls. One ball is taken out without looking. What is the probability that it is not yellow? (a) 11/19 (b) 1/3 (c) 8/19 (d) 12/19
13. 13.What is the smallest number of children in a group that makes sure at least three of them were born in the same month? (a) 36 (b) 24 (c) 25 (d) 37
14. 14.Two straight lines cross. One of the angles is 75°. What is the angle next to it on the same straight line? (a) 15° (b) 75° (c) 285° (d) 105°
15. 15.The numbers 1 to 9 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 5 (b) 15 (c) 45 (d) 18
16. 16.A bag has 3 black balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 2 (c) 5 (d) 4
17. 17.In a class of 27 children, 11 like tea, 22 like coffee and 8 like both. How many like neither? (a) 8 (b) 3 (c) 10 (d) 2
18. 18.What is the prime factorisation of 20? (a) 2³ × 5 (b) 2² × 5 (c) 2 × 5 (d) 2 × 5²
19. 19.A book has 200 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 492 (b) 483 (c) 400 (d) 493
20. 20.In a class of 37 children, 9 like tea, 21 like coffee and 2 like both. How many like at least one of the two? (a) 26 (b) 30 (c) 28 (d) 37
21. 21.A drawer has lots of red, white, black and blue 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) 4 (c) 8 (d) 2
22. 22.Two angles of a triangle are 25° and 75°. What is the third angle? (a) 80° (b) 105° (c) 155° (d) 100°
23. 23.How many two-digit numbers have digits that add up to 7? (a) 6 (b) 7 (c) 10 (d) 8
24. 24.Two straight lines cross. One of the angles is 35°. What is the angle next to it on the same straight line? (a) 325° (b) 55° (c) 145° (d) 35°
25. 25.Two straight lines cross. Two vertically opposite angles add up to 150°. What is each of the other two angles? (a) 30° (b) 210° (c) 105° (d) 75°
26. 26.Two dice are rolled together. What is the probability that the two numbers add up to 5? (a) 1/9 (b) 4/11 (c) 5/36 (d) 1/11
27. 27.What is the circumference of a circle of diameter 42 cm? (Use π = 22/7.) (a) 66 cm (b) 132 cm (c) 1386 cm (d) 264 cm
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
29. 29.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 9 (d) 20
30. 30.The numbers 14 to 22 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 18 (b) 57 (c) 162 (d) 54
31. 31.A canteen sells 3 kinds of sandwich and 3 kinds of juice. In how many ways can you choose one sandwich and one juice? (a) 6 (b) 12 (c) 9 (d) 18
32. 32.What is the mode of 12, 11, 20, 10, 10, 20 and 20? (a) 20 (b) 12 (c) 10 (d) 11
33. 33.If the side of a square is made 4 times as long, how many times as big does its area become? (a) 4 times (b) 8 times (c) 64 times (d) 16 times
34. 34.Which term of the AP 5, 10, 15, 20, … is 120? (a) 24th (b) 25th (c) 23rd (d) 26th
35. 35.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) 14 (c) 15 (d) 7
36. 36.A bag has 6 yellow balls and 5 blue 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) 7 (c) 11 (d) 2
37. 37.How many factors does 18 have? (a) 6 (b) 5 (c) 7 (d) 4
38. 38.Two angles of a triangle are 75° and 75°. What is the exterior angle at the third corner? (a) 210° (b) 30° (c) 105° (d) 150°
39. 39.Two angles of a triangle are 80° and 25°. What is the third angle? (a) 105° (b) 75° (c) 100° (d) 155°
40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16

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