Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 531 · 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 book has 99 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 189 (b) 180 (c) 190 (d) 198
2. 2.In a class of 40 children, 16 like dance, 18 like music and 8 like neither. How many like both? (a) 32 (b) 2 (c) 3 (d) 8
3. 3.What is the sum of the first 11 terms of the AP 4, 7, 10, …? (a) 34 (b) 418 (c) 212 (d) 209
4. 4.What is the prime factorisation of 35? (a) 5³ × 7 (b) 5² × 7 (c) 5 × 7² (d) 5 × 7
5. 5.What is the sum of all the whole numbers from 1 to 54? (a) 1485 (b) 1431 (c) 1458 (d) 2970
6. 6.What is the 32nd term of the AP 5, 14, 23, 32, …? (a) 284 (b) 293 (c) 279 (d) 288
7. 7.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4925 (c) 4225 (d) 5635
8. 8.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) 29 (b) 28 (c) 22 (d) 21
9. 9.A bag has 12 black balls and 13 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 14 (b) 2 (c) 13 (d) 25
10. 10.What is the smallest prime factor of 341? (a) 1 (b) 31 (c) 11 (d) 2
11. 11.What is the mean of 20, 3, 15 and 18? (a) 13 (b) 15 (c) 16 (d) 14
12. 12.What is the midpoint of the line joining (3, 5) and (13, 1)? (a) (3, 8) (b) (5, 2) (c) (8, 3) (d) (16, 6)
13. 13.What is the midpoint of the line joining (9, 3) and (17, 13)? (a) (13, 8) (b) (4, 5) (c) (26, 16) (d) (8, 13)
14. 14.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2411 (d) 2601
15. 15.A bag has 9 green balls and 6 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 2 (c) 15 (d) 11
16. 16.Two straight lines cross. Two vertically opposite angles add up to 250°. What is each of the other two angles? (a) 125° (b) 56° (c) 55° (d) 110°
17. 17.In a league, 13 teams each play every other team once. How many matches are played? (a) 78 (b) 156 (c) 91 (d) 169
18. 18.What is the mode of 5, 1, 5, 7, 1, 1 and 10? (a) 10 (b) 7 (c) 1 (d) 5
19. 19.How many two-digit numbers have digits that add up to 6? (a) 7 (b) 6 (c) 10 (d) 5
20. 20.How many prime numbers are there between 40 and 70? (a) 6 (b) 7 (c) 8 (d) 9
21. 21.What is the prime factorisation of 140? (a) 2³ × 5 × 7 (b) 2 × 5 × 7 (c) 2 × 5 × 7² (d) 2² × 5 × 7
22. 22.6 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 15 (b) 21 (c) 30 (d) 36
23. 23.Two dice are rolled together. What is the probability that the two numbers add up to 5? (a) 1/11 (b) 4/11 (c) 5/36 (d) 1/9
24. 24.A fair dice is rolled once. What is the probability of getting a prime number? (a) 1/2 (b) 1/3 (c) 2/3 (d) 1/6
25. 25.Two angles of a triangle are 55° and 40°. What is the exterior angle at the third corner? (a) 125° (b) 85° (c) 95° (d) 140°
26. 26.Which term of the AP 18, 20, 22, 24, … is 94? (a) 38th (b) 47th (c) 40th (d) 39th
27. 27.Two coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/3 (b) 1/4 (c) 2/3 (d) 1/2
28. 28.In an isosceles triangle, the two equal angles are 45° each. What is the third angle? (a) 91° (b) 135° (c) 90° (d) 45°
29. 29.In how many different ways can the letters of the word SUN be arranged? (a) 3 (b) 6 (c) 9 (d) 12
30. 30.A bag has 6 yellow, 3 white and 3 blue balls. One ball is taken out without looking. What is the probability that it is yellow? (a) 1/12 (b) 1/2 (c) 1/3 (d) 7/12
31. 31.In a class of 41 children, 14 like cricket, 28 like football and 7 like neither. How many like both? (a) 1 (b) 34 (c) 8 (d) 7
32. 32.What is the smallest prime factor of 377? (a) 29 (b) 13 (c) 2 (d) 1
33. 33.In an isosceles triangle, the angle between the two equal sides is 150°. What is each of the other two angles? (a) 25° (b) 30° (c) 15° (d) 150°
34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 376 (d) 196
35. 35.A bag has 7 blue, 7 green and 8 white balls. One ball is taken out without looking. What is the probability that it is not green? (a) 8/11 (b) 15/22 (c) 7/22 (d) 1/3
36. 36.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) 8 (b) 15 (c) 14 (d) 7
37. 37.What is the area of a circle of diameter 42 cm? (Use π = 22/7.) (a) 693 sq cm (b) 5544 sq cm (c) 132 sq cm (d) 1386 sq cm
38. 38.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
39. 39.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
40. 40.How far is the point (3, 4) from the origin (0, 0)? (a) 5 (b) 7 (c) 6 (d) 25

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