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Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 576 · 40 questions
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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.

NameDateTime takenScore ___ / 40
  1. 1.13 friends meet and each one shakes hands once with every other friend. How many handshakes are there? (a) 156 (b) 169 (c) 91 (d) 78
  2. 2.In an isosceles triangle, the angle between the two equal sides is 40°. What is each of the other two angles? (a) 70° (b) 40° (c) 140° (d) 50°
  3. 3.What is the sum of all the whole numbers from 1 to 35? (a) 595 (b) 631 (c) 1260 (d) 630
  4. 4.In an isosceles triangle, the angle between the two equal sides is 150°. What is each of the other two angles? (a) 25° (b) 15° (c) 30° (d) 150°
  5. 5.In a class of 44 children, 24 like dance, 26 like music and 6 like neither. How many like both? (a) 12 (b) 38 (c) 13 (d) 6
  6. 6.What is the smallest prime factor of 145? (a) 29 (b) 2 (c) 1 (d) 5
  7. 7.In an isosceles triangle, the two equal angles are 70° each. What is the third angle? (a) 40° (b) 70° (c) 20° (d) 110°
  8. 8.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
  9. 9.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  10. 10.What is the mean of 7, 10, 17 and 22? (a) 15 (b) 13 (c) 16 (d) 14
  11. 11.If every edge of a cube is made 4 times as long, how many times as big does its volume become? (a) 64 times (b) 4 times (c) 8 times (d) 16 times
  12. 12.What is the mean of 4, 19, 20, 4 and 8? (a) 13 (b) 10 (c) 12 (d) 11
  13. 13.In an isosceles triangle, the angle between the two equal sides is 130°. What is each of the other two angles? (a) 130° (b) 35° (c) 25° (d) 50°
  14. 14.In an isosceles triangle, the two equal angles are 45° each. What is the third angle? (a) 91° (b) 90° (c) 45° (d) 135°
  15. 15.What is the 10th term of the AP 13, 17, 21, 25, …? (a) 53 (b) 49 (c) 40 (d) 36
  16. 16.Two angles of a triangle are 35° and 20°. What is the third angle? (a) 55° (b) 125° (c) 160° (d) 145°
  17. 17.Two angles of a triangle are 30° and 50°. What is the third angle? (a) 150° (b) 80° (c) 130° (d) 100°
  18. 18.A bag has 3 black balls and 11 blue 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) 12 (d) 4
  19. 19.The numbers 7 to 15 are placed in a 3 × 3 magic square, so every row, column and diagonal has the same sum. What is that sum? (a) 11 (b) 36 (c) 33 (d) 99
  20. 20.In a class of 40 children, 15 like cricket, 21 like football and 3 like both. How many like neither? (a) 4 (b) 7 (c) 10 (d) 3
  21. 21.Two straight lines cross. Two vertically opposite angles add up to 310°. What is each of the other two angles? (a) 26° (b) 25° (c) 50° (d) 155°
  22. 22.What is the smallest prime factor of 217? (a) 2 (b) 31 (c) 7 (d) 1
  23. 23.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) 7 (b) 15 (c) 14 (d) 8
  24. 24.How many two-digit numbers have digits that add up to 7? (a) 10 (b) 6 (c) 7 (d) 8
  25. 25.What is the sum of the first 10 terms of the AP 1, 2, 3, …? (a) 55 (b) 60 (c) 110 (d) 10
  26. 26.A bag has 8 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 11 (c) 9 (d) 2
  27. 27.What is the smallest prime factor of 77? (a) 2 (b) 11 (c) 7 (d) 1
  28. 28.Two straight lines cross. Two vertically opposite angles add up to 140°. What is each of the other two angles? (a) 70° (b) 110° (c) 40° (d) 220°
  29. 29.In how many different ways can the letters of the word PEN be arranged? (a) 9 (b) 3 (c) 6 (d) 12
  30. 30.Two angles of a triangle are 80° and 85°. What is the third angle? (a) 100° (b) 15° (c) 95° (d) 165°
  31. 31.A bag has 3 red balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 5 (b) 2 (c) 6 (d) 4
  32. 32.What is the sum of the first 16 odd numbers (1 + 3 + 5 + …)? (a) 136 (b) 272 (c) 256 (d) 31
  33. 33.In an isosceles triangle, the two equal angles are 30° each. What is the third angle? (a) 120° (b) 60° (c) 150° (d) 30°
  34. 34.How many two-digit numbers have digits that add up to 17? (a) 1 (b) 3 (c) 2 (d) 18
  35. 35.What is the midpoint of the line joining (10, 10) and (22, 16)? (a) (16, 13) (b) (32, 26) (c) (13, 16) (d) (6, 3)
  36. 36.A bag has 10 yellow balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 16 (b) 11 (c) 2 (d) 7
  37. 37.How far is the point (16, 12) from the origin (0, 0)? (a) 20 (b) 28 (c) 400 (d) 21
  38. 38.If the side of a square is made 2 times as long, how many times as big does its area become? (a) 4 times (b) 5 times (c) 8 times (d) 2 times
  39. 39.How many two-digit numbers have digits that add up to 6? (a) 5 (b) 6 (c) 10 (d) 7
  40. 40.What is the sum of the first 10 terms of the AP 2, 5, 8, …? (a) 155 (b) 29 (c) 170 (d) 310

Answer key

  1. 78
  2. 70°
  3. 630
  4. 15°
  5. 12
  6. 5
  7. 40°
  8. 21
  9. (x + 6)(x + 7)
  10. 14
  11. 64 times
  12. 11
  13. 25°
  14. 90°
  15. 49
  16. 125°
  17. 100°
  18. 12
  19. 33
  20. 7
  21. 25°
  22. 7
  23. 8
  24. 7
  25. 55
  26. 9
  27. 7
  28. 110°
  29. 6
  30. 15°
  31. 5
  32. 256
  33. 120°
  34. 2
  35. (16, 13)
  36. 11
  37. 20
  38. 4 times
  39. 6
  40. 155

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