Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 708 · 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 an isosceles triangle, the two equal angles are 70° each. What is the third angle? (a) 70° (b) 20° (c) 110° (d) 40°
2. 2.How many two-digit numbers have digits that add up to 12? (a) 7 (b) 8 (c) 6 (d) 13
3. 3.How many times is the digit 1 written when you write all the numbers from 1 to 79? (a) 7 (b) 17 (c) 18 (d) 19
4. 4.A book has 39 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 70 (b) 78 (c) 69 (d) 60
5. 5.How many two-digit numbers have digits that add up to 13? (a) 6 (b) 14 (c) 5 (d) 7
6. 6.Two angles of a triangle are 20° and 50°. What is the third angle? (a) 160° (b) 70° (c) 110° (d) 130°
7. 7.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
8. 8.A cuboid is 9 cm long, 8 cm wide and 2 cm high. What is its total surface area? (a) 106 sq cm (b) 424 sq cm (c) 212 sq cm (d) 144 sq cm
9. 9.In an isosceles triangle, the angle between the two equal sides is 70°. What is each of the other two angles? (a) 20° (b) 70° (c) 55° (d) 110°
10. 10.What is the midpoint of the line joining (2, 6) and (4, 12)? (a) (3, 9) (b) (1, 3) (c) (9, 3) (d) (6, 18)
11. 11.What is the midpoint of the line joining (3, 8) and (5, 12)? (a) (4, 10) (b) (8, 20) (c) (10, 4) (d) (1, 2)
12. 12.How many two-digit numbers have digits that add up to 9? (a) 11 (b) 8 (c) 10 (d) 9
13. 13.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 88 (c) 52 (d) 40
14. 14.How many times is the digit 1 written when you write all the numbers from 1 to 64? (a) 6 (b) 16 (c) 18 (d) 17
15. 15.What is the mode of 4, 4, 16, 11, 4, 14 and 11? (a) 4 (b) 16 (c) 14 (d) 11
16. 16.Which term of the AP 5, 9, 13, 17, … is 113? (a) 27th (b) 29th (c) 30th (d) 28th
17. 17.A team of 2 players is to be chosen from 8 players. How many different teams are possible? (a) 16 (b) 56 (c) 28 (d) 64
18. 18.The numbers 9 to 17 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 117 (b) 13 (c) 39 (d) 42
19. 19.How many prime numbers are there between 30 and 50? (a) 6 (b) 4 (c) 7 (d) 5
20. 20.A bag has 4 black balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 2 (c) 7 (d) 5
21. 21.The mean of 3 numbers is 21. When the number 33 is added to them, what is the new mean? (a) 25 (b) 24 (c) 21 (d) 27
22. 22.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 16 (d) 10
23. 23.7 T-shirts and 3 pairs of shorts are in a cupboard. How many different outfits of one T-shirt and one pair of shorts can be made? (a) 10 (b) 28 (c) 21 (d) 42
24. 24.In a class of 41 children, 24 like maths, 23 like science and 12 like both. How many like only maths? (a) 18 (b) 12 (c) 11 (d) 24
25. 25.The mean of 4 numbers is 27. When the number 7 is added to them, what is the new mean? (a) 24 (b) 17 (c) 23 (d) 27
26. 26.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4624 (d) 4896
27. 27.What is the sum of all the whole numbers from 1 to 92? (a) 4278 (b) 8556 (c) 4232 (d) 4186
28. 28.In a class of 43 children, 21 like maths, 24 like science and 12 like both. How many like only maths? (a) 19 (b) 21 (c) 12 (d) 9
29. 29.What is the sum of the first 7 terms of the AP 10, 11, 12, …? (a) 182 (b) 16 (c) 92 (d) 91
30. 30.How many two-digit numbers have digits that add up to 3? (a) 4 (b) 10 (c) 2 (d) 3
31. 31.What is the distance between the points (2, 8) and (6, 11)? (a) 7 (b) 25 (c) 5 (d) 6
32. 32.What is the midpoint of the line joining (8, 17) and (20, 7)? (a) (28, 24) (b) (14, 12) (c) (6, 5) (d) (12, 14)
33. 33.What is the prime factorisation of 112? (a) 2⁴ × 7 (b) 2 × 7⁴ (c) 2⁵ × 7 (d) 2³ × 7
34. 34.How far is the point (12, 16) from the origin (0, 0)? (a) 400 (b) 28 (c) 20 (d) 21
35. 35.How far is the point (6, 8) from the origin (0, 0)? (a) 14 (b) 11 (c) 100 (d) 10
36. 36.8 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 56 (b) 36 (c) 28 (d) 64
37. 37.A drawer has lots of 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) 4 (b) 2 (c) 5 (d) 3
38. 38.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 4 (c) 196 (d) 100
39. 39.What is the midpoint of the line joining (3, 5) and (15, 1)? (a) (3, 9) (b) (9, 3) (c) (6, 2) (d) (18, 6)
40. 40.What is the sum of the first 9 terms of the AP 9, 11, 13, …? (a) 25 (b) 153 (c) 306 (d) 162

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