Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 318 · 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.The numbers 18 to 26 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 22 (b) 198 (c) 69 (d) 66
2. 2.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 13 (d) 19
3. 3.Two angles of a triangle are 30° and 50°. What is the exterior angle at the third corner? (a) 100° (b) 80° (c) 130° (d) 150°
4. 4.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
5. 5.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 10609 (c) 9409 (d) 9991
6. 6.A bag has 4 black balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 2 (c) 12 (d) 5
7. 7.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
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 month? (a) 36 (b) 37 (c) 48 (d) 49
9. 9.A bag has 5 green balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 7 (c) 6 (d) 2
10. 10.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
11. 11.What is the mean of 18, 16, 6 and 4? (a) 11 (b) 13 (c) 10 (d) 12
12. 12.What is the midpoint of the line joining (2, 10) and (4, 16)? (a) (1, 3) (b) (13, 3) (c) (6, 26) (d) (3, 13)
13. 13.How many prime numbers are there between 60 and 80? (a) 4 (b) 6 (c) 5 (d) 7
14. 14.Two coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/2 (b) 1/3 (c) 1/4 (d) 2/3
15. 15.Which term of the AP 16, 18, 20, 22, … is 72? (a) 36th (b) 29th (c) 30th (d) 28th
16. 16.In an isosceles triangle, the angle between the two equal sides is 30°. What is each of the other two angles? (a) 150° (b) 30° (c) 60° (d) 75°
17. 17.In a class of 38 children, 22 like tea, 12 like coffee and 6 like both. How many like neither? (a) 4 (b) 16 (c) 6 (d) 10
18. 18.In how many different ways can the letters of the word CHAIR be arranged? (a) 20 (b) 240 (c) 120 (d) 25
19. 19.What is the mode of 2, 17, 12, 2, 17, 9 and 17? (a) 2 (b) 17 (c) 9 (d) 12
20. 20.What is the area of a circle of radius 28 cm? (Use π = 22/7.) (a) 1232 sq cm (b) 9856 sq cm (c) 176 sq cm (d) 2464 sq cm
21. 21.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
22. 22.How many two-digit numbers have digits that add up to 5? (a) 6 (b) 10 (c) 5 (d) 4
23. 23.What is the sum of the first 20 odd numbers (1 + 3 + 5 + …)? (a) 420 (b) 39 (c) 400 (d) 210
24. 24.What is the smallest prime factor of 87? (a) 2 (b) 29 (c) 1 (d) 3
25. 25.9 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 81 (b) 72 (c) 36 (d) 45
26. 26.How far is the point (8, 15) from the origin (0, 0)? (a) 17 (b) 289 (c) 18 (d) 23
27. 27.How many times is the digit 9 written when you write all the numbers from 1 to 89? (a) 8 (b) 9 (c) 7 (d) 10
28. 28.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) 54 (b) 18 (c) 21 (d) 6
29. 29.Three coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/4 (b) 1/8 (c) 3/8 (d) 3/4
30. 30.Two dice are rolled together. What is the probability that the two numbers add up to 8? (a) 1/11 (b) 1/6 (c) 5/11 (d) 5/36
31. 31.Which term of the AP 19, 24, 29, 34, … is 69? (a) 10th (b) 11th (c) 13th (d) 12th
32. 32.What is the mean of 3, 7, 20, 18 and 7? (a) 10 (b) 11 (c) 13 (d) 12
33. 33.How many times is the digit 2 written when you write all the numbers from 1 to 36? (a) 3 (b) 14 (c) 13 (d) 15
34. 34.The mean of 6 numbers is 40. When the number 33 is added to them, what is the new mean? (a) 39 (b) 41 (c) 38 (d) 40
35. 35.What is the midpoint of the line joining (7, 2) and (9, 6)? (a) (1, 2) (b) (16, 8) (c) (4, 8) (d) (8, 4)
36. 36.A captain and a vice-captain are to be chosen from 11 players. In how many ways can this be done? (a) 55 (b) 22 (c) 110 (d) 121
37. 37.Three coins are tossed together. What is the probability of getting at least one head? (a) 1/4 (b) 1/8 (c) 3/8 (d) 7/8
38. 38.What is the mode of 5, 18, 18, 5, 18, 8 and 14? (a) 8 (b) 14 (c) 5 (d) 18
39. 39.A bag has 8 red balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 9 (b) 7 (c) 2 (d) 14
40. 40.What is the sum of the first 22 odd numbers (1 + 3 + 5 + …)? (a) 253 (b) 43 (c) 506 (d) 484

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