Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 25 · 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.How many times is the digit 2 written when you write all the numbers from 1 to 36? (a) 13 (b) 15 (c) 14 (d) 3
2. 2.What is the sum of the first 10 terms of the AP 6, 8, 10, …? (a) 150 (b) 300 (c) 24 (d) 160
3. 3.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 1156 (d) 916
4. 4.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
5. 5.What is the smallest prime factor of 155? (a) 5 (b) 1 (c) 2 (d) 31
6. 6.What is the mean of 18, 10, 20 and 8? (a) 13 (b) 15 (c) 14 (d) 16
7. 7.There are 7 roads from town A to town B and 3 roads from town B to town C. How many different routes go from A to C through B? (a) 21 (b) 10 (c) 42 (d) 28
8. 8.Two straight lines cross. One of the angles is 85°. What is the angle next to it on the same straight line? (a) 275° (b) 85° (c) 5° (d) 95°
9. 9.How many times is the digit 8 written when you write all the numbers from 1 to 41? (a) 4 (b) 2 (c) 5 (d) 3
10. 10.The numbers 5 to 13 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 81 (b) 9 (c) 30 (d) 27
11. 11.How many times is the digit 3 written when you write all the numbers from 1 to 30? (a) 4 (b) 3 (c) 2 (d) 5
12. 12.What is the missing term in the AP 19, ?, 29, 34, 39? (a) 29 (b) 24 (c) 23 (d) 25
13. 13.What is the distance between the points (2, 5) and (9, 29)? (a) 625 (b) 31 (c) 25 (d) 26
14. 14.How many factors does 58 have? (a) 4 (b) 5 (c) 3 (d) 2
15. 15.What is the sum of the first 26 odd numbers (1 + 3 + 5 + …)? (a) 351 (b) 51 (c) 702 (d) 676
16. 16.A cuboid is 12 cm long, 10 cm wide and 4 cm high. What is its volume? (a) 416 cu cm (b) 120 cu cm (c) 480 cu cm (d) 26 cu cm
17. 17.In which quadrant does the point (−4, −4) lie? (a) Second quadrant (b) First quadrant (c) Fourth quadrant (d) Third quadrant
18. 18.10 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 100 (b) 45 (c) 90 (d) 55
19. 19.What is the midpoint of the line joining (10, 2) and (22, 12)? (a) (32, 14) (b) (6, 5) (c) (7, 16) (d) (16, 7)
20. 20.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 6 (c) 180 (d) 324
21. 21.A fair dice is rolled once. What is the probability of getting a number greater than 1? (a) 1/3 (b) 5/6 (c) 1/5 (d) 1/6
22. 22.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 81 (d) 1
23. 23.A drawer has lots of black, 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) 6 (b) 2 (c) 4 (d) 3
24. 24.Two dice are rolled together. What is the probability that the two numbers add up to 8? (a) 5/11 (b) 1/6 (c) 5/36 (d) 1/11
25. 25.What is the mode of 3, 13, 8, 3, 13, 13 and 11? (a) 8 (b) 3 (c) 11 (d) 13
26. 26.The numbers 3 to 11 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) 21 (c) 63 (d) 7
27. 27.In a class of 46 children, 18 like maths, 26 like science and 6 like both. How many like only maths? (a) 6 (b) 12 (c) 18 (d) 20
28. 28.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 26 (d) 28
29. 29.What is the median of 1, 33, 37, 23, 32, 6 and 15? (a) 23 (b) 21 (c) 32 (d) 15
30. 30.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 12 (d) 42
31. 31.A team of 2 players is to be chosen from 9 players. How many different teams are possible? (a) 72 (b) 81 (c) 36 (d) 18
32. 32.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 35) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
33. 33.In a league, 13 teams each play every other team once. How many matches are played? (a) 169 (b) 91 (c) 156 (d) 78
34. 34.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 6 (d) 5
35. 35.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 3 and 6 (c) 2 and 7 (d) 1 and 8
36. 36.What is the prime factorisation of 35? (a) 5³ × 7 (b) 5 × 7² (c) 5² × 7 (d) 5 × 7
37. 37.In how many different ways can the letters of the word STAR be arranged? (a) 24 (b) 12 (c) 48 (d) 16
38. 38.Two angles of a triangle are 30° and 40°. What is the exterior angle at the third corner? (a) 150° (b) 70° (c) 110° (d) 140°
39. 39.A book has 25 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 41 (b) 42 (c) 50 (d) 32
40. 40.In a class of 40 children, 17 like tea, 18 like coffee and 5 like both. How many like only tea? (a) 22 (b) 17 (c) 12 (d) 13

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