Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 493 · 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.What is the midpoint of the line joining (5, 9) and (11, 19)? (a) (16, 28) (b) (3, 5) (c) (8, 14) (d) (14, 8)
2. 2.A cube has edges of 12 cm. What is its total surface area? (a) 864 sq cm (b) 1728 sq cm (c) 144 sq cm (d) 576 sq cm
3. 3.In an isosceles triangle, the angle between the two equal sides is 30°. What is each of the other two angles? (a) 75° (b) 60° (c) 150° (d) 30°
4. 4.The numbers 8 to 16 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 12 (b) 108 (c) 36 (d) 39
5. 5.What is the product of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 17 (d) 19
6. 6.What is the median of 29, 6, 8, 19, 24, 3 and 33? (a) 20 (b) 8 (c) 19 (d) 24
7. 7.What is the midpoint of the line joining (7, 16) and (19, 10)? (a) (6, 3) (b) (14, 13) (c) (13, 13) (d) (26, 26)
8. 8.A drawer has lots of yellow, blue 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) 3 (d) 4
9. 9.In a class of 40 children, 19 like dance, 24 like music and 10 like both. How many like only dance? (a) 14 (b) 19 (c) 9 (d) 16
10. 10.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) 14 (b) 15 (c) 8 (d) 7
11. 11.A canteen sells 4 kinds of sandwich and 3 kinds of juice. In how many ways can you choose one sandwich and one juice? (a) 24 (b) 7 (c) 16 (d) 12
12. 12.A bag has 11 blue balls and 7 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 18 (c) 13 (d) 12
13. 13.What is the sum of the first 9 odd numbers (1 + 3 + 5 + …)? (a) 17 (b) 45 (c) 90 (d) 81
14. 14.What is the smallest prime factor of 65? (a) 5 (b) 2 (c) 13 (d) 1
15. 15.What is the prime factorisation of 21? (a) 3 × 7 (b) 3 × 7² (c) 3³ × 7 (d) 3² × 7
16. 16.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) 21 (c) 22 (d) 14
17. 17.How many factors does 97 have? (a) 1 (b) 3 (c) 4 (d) 2
18. 18.A drawer has lots of green, blue, yellow and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 8 (c) 4 (d) 5
19. 19.Two angles of a triangle are 30° and 120°. What is the exterior angle at the third corner? (a) 210° (b) 60° (c) 30° (d) 150°
20. 20.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
21. 21.In a class of 26 children, 10 like dance, 8 like music and 2 like both. How many like neither? (a) 2 (b) 12 (c) 10 (d) 8
22. 22.A bag has 5 yellow balls and 3 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) 6 (c) 4 (d) 8
23. 23.Two angles of a triangle are 65° and 40°. What is the third angle? (a) 105° (b) 115° (c) 75° (d) 140°
24. 24.In a class of 27 children, 12 like dance, 12 like music and 9 like neither. How many like both? (a) 9 (b) 7 (c) 18 (d) 6
25. 25.What is the mode of 17, 9, 15, 16, 16, 16 and 15? (a) 9 (b) 17 (c) 16 (d) 15
26. 26.How many times is the digit 5 written when you write all the numbers from 1 to 96? (a) 20 (b) 9 (c) 21 (d) 19
27. 27.In a class of 35 children, 12 like dance, 20 like music and 5 like both. How many like neither? (a) 13 (b) 3 (c) 5 (d) 8
28. 28.How far is the point (15, 8) from the origin (0, 0)? (a) 23 (b) 289 (c) 18 (d) 17
29. 29.Two straight lines cross. One of the angles is 165°. What is the angle next to it on the same straight line? (a) 15° (b) 195° (c) 165° (d) 16°
30. 30.How many prime numbers are there between 70 and 100? (a) 7 (b) 6 (c) 5 (d) 8
31. 31.In a league, 9 teams each play every other team once. How many matches are played? (a) 36 (b) 45 (c) 72 (d) 81
32. 32.What is the sum of the first 10 terms of the AP 1, 2, 3, …? (a) 55 (b) 110 (c) 60 (d) 10
33. 33.Two angles of a triangle are 45° and 105°. What is the exterior angle at the third corner? (a) 75° (b) 135° (c) 150° (d) 30°
34. 34.The mean of 8 numbers is 35. When the number 44 is added to them, what is the new mean? (a) 36 (b) 37 (c) 38 (d) 35
35. 35.What is the circumference of a circle of diameter 14 cm? (Use π = 22/7.) (a) 88 cm (b) 44 cm (c) 22 cm (d) 154 cm
36. 36.There are 6 roads from town A to town B and 7 roads from town B to town C. How many different routes go from A to C through B? (a) 13 (b) 84 (c) 42 (d) 48
37. 37.What is the mode of 7, 7, 4, 12, 7, 13 and 4? (a) 13 (b) 12 (c) 7 (d) 4
38. 38.Two dice are rolled together. What is the probability that the two numbers add up to 4? (a) 1/9 (b) 3/11 (c) 1/11 (d) 1/12
39. 39.A fair dice is rolled once. What is the probability of getting a number greater than 3? (a) 1/3 (b) 1/2 (c) 2/3 (d) 1/6
40. 40.How far is the point (4, 3) from the origin (0, 0)? (a) 5 (b) 25 (c) 7 (d) 6

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