Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 899 · 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 bag has 10 red balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 11 (c) 12 (d) 15
2. 2.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10009 (d) 10619
3. 3.What is the midpoint of the line joining (9, 9) and (11, 13)? (a) (11, 10) (b) (10, 11) (c) (20, 22) (d) (1, 2)
4. 4.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
5. 5.4 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) 56 (b) 11 (c) 28 (d) 32
6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 541 (d) 361
7. 7.How many two-digit numbers have digits that add up to 4? (a) 10 (b) 4 (c) 5 (d) 3
8. 8.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
9. 9.Two dice are rolled together. What is the probability that the two numbers add up to 8? (a) 1/6 (b) 1/11 (c) 5/11 (d) 5/36
10. 10.Two coins are tossed together. What is the probability of getting exactly one head? (a) 2/3 (b) 1/3 (c) 1/4 (d) 1/2
11. 11.What is the median of 9, 2, 17, 11 and 32? (a) 11 (b) 17 (c) 12 (d) 9
12. 12.What is the median of 6, 30, 12, 8, 26, 18 and 1? (a) 12 (b) 8 (c) 13 (d) 18
13. 13.Two dice are rolled together. What is the probability that the two numbers add up to 10? (a) 1/9 (b) 3/11 (c) 1/11 (d) 1/12
14. 14.What is the mean of 10, 8, 16 and 14? (a) 16 (b) 13 (c) 11 (d) 12
15. 15.Two angles of a triangle are 40° and 90°. What is the third angle? (a) 130° (b) 140° (c) 90° (d) 50°
16. 16.How far is the point (6, 8) from the origin (0, 0)? (a) 100 (b) 11 (c) 14 (d) 10
17. 17.What is the mean of 13, 15, 2 and 6? (a) 10 (b) 12 (c) 8 (d) 9
18. 18.A bag has 8 black 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) 10 (c) 15 (d) 9
19. 19.Two straight lines cross. Two vertically opposite angles add up to 50°. What is each of the other two angles? (a) 310° (b) 25° (c) 130° (d) 155°
20. 20.If the side of a square is made 5 times as long, how many times as big does its area become? (a) 10 times (b) 5 times (c) 25 times (d) 125 times
21. 21.5 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 20 (b) 10 (c) 15 (d) 25
22. 22.Which term of the AP 9, 11, 13, 15, … is 67? (a) 31st (b) 32nd (c) 30th (d) 29th
23. 23.A drawer has lots of red, blue, yellow and white 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) 5 (d) 4
24. 24.If the radius of a circle is made 4 times as long, how many times as big does its area become? (a) 8 times (b) 16 times (c) 64 times (d) 4 times
25. 25.In an isosceles triangle, the angle between the two equal sides is 100°. What is each of the other two angles? (a) 80° (b) 100° (c) 40° (d) 50°
26. 26.15 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 210 (b) 120 (c) 225 (d) 105
27. 27.How many factors does 52 have? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.A drawer has lots of white, 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) 2 (b) 6 (c) 3 (d) 4
29. 29.A bag has 10 red balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 15 (c) 2 (d) 11
30. 30.A fair dice is rolled once. What is the probability of getting a prime number? (a) 1/3 (b) 2/3 (c) 1/6 (d) 1/2
31. 31.What is the sum of the first 12 terms of the AP 9, 14, 19, …? (a) 468 (b) 438 (c) 876 (d) 64
32. 32.How far is the point (15, 8) from the origin (0, 0)? (a) 17 (b) 18 (c) 289 (d) 23
33. 33.Two angles of a triangle are 75° and 30°. What is the exterior angle at the third corner? (a) 75° (b) 150° (c) 255° (d) 105°
34. 34.A team of 2 players is to be chosen from 12 players. How many different teams are possible? (a) 144 (b) 66 (c) 132 (d) 24
35. 35.If the side of a square is made 2 times as long, how many times as big does its area become? (a) 5 times (b) 2 times (c) 4 times (d) 8 times
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
37. 37.A book has 202 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 498 (b) 499 (c) 489 (d) 404
38. 38.What is the mode of 19, 19, 6, 7, 5, 6 and 6? (a) 6 (b) 7 (c) 5 (d) 19
39. 39.What is the mean of 17, 18, 5, 15 and 20? (a) 17 (b) 14 (c) 16 (d) 15
40. 40.What is the prime factorisation of 40? (a) 2 × 5³ (b) 2² × 5 (c) 2³ × 5 (d) 2⁴ × 5

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