Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 19 · 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 coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/3 (b) 1/2 (c) 2/3 (d) 1/4
2. 2.How many two-digit numbers have digits that add up to 5? (a) 10 (b) 6 (c) 5 (d) 4
3. 3.In how many different ways can the letters of the word BOAT be arranged? (a) 16 (b) 12 (c) 48 (d) 24
4. 4.In a class of 56 children, 29 like mangoes, 29 like bananas and 10 like both. How many like only mangoes? (a) 10 (b) 27 (c) 29 (d) 19
5. 5.What is the smallest prime factor of 133? (a) 2 (b) 1 (c) 19 (d) 7
6. 6.What is the midpoint of the line joining (7, 5) and (11, 9)? (a) (18, 14) (b) (9, 7) (c) (2, 2) (d) (7, 9)
7. 7.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 214 (c) 144 (d) 109
8. 8.Two angles of a triangle are 65° and 75°. What is the third angle? (a) 140° (b) 115° (c) 105° (d) 40°
9. 9.A cube has edges of 2 cm. What is its total surface area? (a) 24 sq cm (b) 16 sq cm (c) 4 sq cm (d) 8 sq cm
10. 10.How many times is the digit 1 written when you write all the numbers from 1 to 30? (a) 14 (b) 12 (c) 3 (d) 13
11. 11.Two angles of a triangle are 35° and 25°. What is the exterior angle at the third corner? (a) 155° (b) 120° (c) 60° (d) 145°
12. 12.How many factors does 30 have? (a) 7 (b) 6 (c) 9 (d) 8
13. 13.In an isosceles triangle, the two equal angles are 75° each. What is the third angle? (a) 105° (b) 30° (c) 15° (d) 75°
14. 14.A drawer has lots of red 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) 4 (b) 5 (c) 3 (d) 2
15. 15.A captain and a vice-captain are to be chosen from 8 players. In how many ways can this be done? (a) 28 (b) 56 (c) 64 (d) 16
16. 16.What is the smallest number of children in a group that makes sure at least four of them were born in the same day of the week? (a) 28 (b) 29 (c) 22 (d) 21
17. 17.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
19. 19.How many factors does 98 have? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.Two straight lines cross. Two vertically opposite angles add up to 150°. What is each of the other two angles? (a) 75° (b) 210° (c) 105° (d) 30°
21. 21.What is the sum of the first 5 terms of the AP 2, 7, 12, …? (a) 22 (b) 65 (c) 120 (d) 60
22. 22.A captain and a vice-captain are to be chosen from 7 players. In how many ways can this be done? (a) 49 (b) 14 (c) 42 (d) 21
23. 23.What is the 16th term of the AP 13, 15, 17, 19, …? (a) 32 (b) 30 (c) 43 (d) 45
24. 24.In a class of 29 children, 21 like dance, 19 like music and 12 like both. How many like neither? (a) 13 (b) 12 (c) 2 (d) 1
25. 25.What is the mode of 6, 6, 8, 16, 8, 8 and 10? (a) 10 (b) 6 (c) 16 (d) 8
26. 26.Two angles of a triangle are 55° and 95°. What is the third angle? (a) 150° (b) 125° (c) 30° (d) 85°
27. 27.A book has 156 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 361 (b) 351 (c) 312 (d) 360
28. 28.The numbers 4 to 12 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 24 (b) 27 (c) 8 (d) 72
29. 29.What is the smallest number of children in a group that makes sure at least two of them were born in the same month? (a) 13 (b) 25 (c) 12 (d) 24
30. 30.A book has 26 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 44 (b) 34 (c) 52 (d) 43
31. 31.Two angles of a triangle are 35° and 50°. What is the exterior angle at the third corner? (a) 85° (b) 145° (c) 95° (d) 130°
32. 32.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 36 (c) 6 (d) 196
33. 33.The numbers 2 to 10 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 6 (b) 21 (c) 18 (d) 54
34. 34.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
35. 35.How many factors does 118 have? (a) 5 (b) 4 (c) 2 (d) 3
36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 71 (d) 73
37. 37.A captain and a vice-captain are to be chosen from 12 players. In how many ways can this be done? (a) 132 (b) 24 (c) 144 (d) 66
38. 38.What is the circumference of a circle of diameter 14 cm? (Use π = 22/7.) (a) 22 cm (b) 44 cm (c) 88 cm (d) 154 cm
39. 39.What is the mean of 17, 17, 20 and 14? (a) 18 (b) 17 (c) 19 (d) 16
40. 40.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) 18 (c) 45 (d) 15

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