Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 831 · 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.Two angles of a triangle are 30° and 120°. What is the exterior angle at the third corner? (a) 30° (b) 210° (c) 60° (d) 150°
2. 2.What is the prime factorisation of 80? (a) 2⁴ × 5 (b) 2³ × 5 (c) 2⁵ × 5 (d) 2 × 5⁴
3. 3.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 40 (c) 13 (d) 14
4. 4.Two angles of a triangle are 30° and 35°. What is the exterior angle at the third corner? (a) 115° (b) 145° (c) 65° (d) 150°
5. 5.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
6. 6.What is the smallest prime factor of 39? (a) 1 (b) 13 (c) 2 (d) 3
7. 7.In an isosceles triangle, the two equal angles are 85° each. What is the third angle? (a) 10° (b) 95° (c) 85° (d) 5°
8. 8.What is the circumference of a circle of radius 35 cm? (Use π = 22/7.) (a) 3850 cm (b) 110 cm (c) 220 cm (d) 440 cm
9. 9.If the radius of a circle is made 3 times as long, how many times as big does its area become? (a) 27 times (b) 6 times (c) 9 times (d) 3 times
10. 10.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 6 (c) 8 (d) 7
11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
12. 12.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
13. 13.A drawer has lots of black, white, blue, green 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) 5 (c) 6 (d) 2
14. 14.What is the prime factorisation of 210? (a) 2 × 3 × 5 × 7² (b) 2² × 3 × 5 × 7 (c) 2³ × 3 × 5 × 7 (d) 2 × 3 × 5 × 7
15. 15.Two dice are rolled together. What is the probability that the two numbers add up to 9? (a) 4/11 (b) 5/36 (c) 1/9 (d) 1/11
16. 16.What is the median of 7, 8, 14, 33, 24, 4 and 2? (a) 8 (b) 33 (c) 14 (d) 7
17. 17.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
18. 18.Two straight lines cross. One of the angles is 30°. What is the angle next to it on the same straight line? (a) 60° (b) 30° (c) 330° (d) 150°
19. 19.How many times is the digit 6 written when you write all the numbers from 1 to 81? (a) 8 (b) 19 (c) 17 (d) 18
20. 20.Two dice are rolled together. What is the probability that the two numbers add up to 2? (a) 1/11 (b) 1/12 (c) 1/18 (d) 1/36
21. 21.A cuboid is 9 cm long, 3 cm wide and 4 cm high. What is its volume? (a) 27 cu cm (b) 150 cu cm (c) 16 cu cm (d) 108 cu cm
22. 22.What is the circumference of a circle of diameter 14 cm? (Use π = 22/7.) (a) 22 cm (b) 88 cm (c) 44 cm (d) 154 cm
23. 23.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 4 (c) 400 (d) 208
24. 24.In a class of 35 children, 14 like mangoes, 15 like bananas and 2 like both. How many like at least one of the two? (a) 25 (b) 35 (c) 27 (d) 29
25. 25.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) 22 (b) 21 (c) 15 (d) 14
26. 26.What is the median of 1, 3, 39, 26 and 34? (a) 39 (b) 34 (c) 3 (d) 26
27. 27.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 4899 (d) 5041
28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
29. 29.A drawer has lots of green, 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) 2 (b) 3 (c) 6 (d) 4
30. 30.In how many different ways can the letters of the word FISH be arranged? (a) 24 (b) 12 (c) 48 (d) 16
31. 31.In a class of 39 children, 20 like mangoes, 20 like bananas and 9 like both. How many like at least one of the two? (a) 39 (b) 31 (c) 40 (d) 22
32. 32.The numbers 12 to 20 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 51 (b) 16 (c) 48 (d) 144
33. 33.A drawer has lots of white, black, green, 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) 10 (b) 5 (c) 2 (d) 6
34. 34.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 9
35. 35.Two dice are rolled together. What is the probability that the two numbers add up to 4? (a) 1/12 (b) 1/11 (c) 1/9 (d) 3/11
36. 36.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) 24 (b) 37 (c) 25 (d) 36
37. 37.What is the midpoint of the line joining (2, 19) and (12, 7)? (a) (13, 7) (b) (7, 13) (c) (14, 26) (d) (5, 6)
38. 38.Two angles of a triangle are 45° and 70°. What is the exterior angle at the third corner? (a) 135° (b) 110° (c) 115° (d) 65°
39. 39.What is the mean of 11, 11, 13 and 21? (a) 13 (b) 15 (c) 14 (d) 16
40. 40.How many times is the digit 9 written when you write all the numbers from 1 to 73? (a) 6 (b) 8 (c) 7 (d) 5

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