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Maths Olympiad Sample Paper 2 (Practice)

Maths Olympiad topic 33: Maths Olympiad mock test 2 · Set 180 · 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.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 90 (d) 117
  2. 2.What is the prime factorisation of 84? (a) 2² × 3 × 7 (b) 2 × 3 × 7² (c) 2³ × 3 × 7 (d) 2 × 3 × 7
  3. 3.A captain and a vice-captain are to be chosen from 10 players. In how many ways can this be done? (a) 20 (b) 100 (c) 45 (d) 90
  4. 4.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 81 (d) 9
  5. 5.What is the median of 12, 22, 35, 6, 24, 7 and 13? (a) 6 (b) 13 (c) 17 (d) 22
  6. 6.Three coins are tossed together. What is the probability of getting at least one head and at least one tail? (a) 1/4 (b) 3/4 (c) 1/8 (d) 3/8
  7. 7.What is the sum of the first 12 odd numbers (1 + 3 + 5 + …)? (a) 144 (b) 156 (c) 23 (d) 78
  8. 8.How many times is the digit 6 written when you write all the numbers from 1 to 63? (a) 10 (b) 11 (c) 8 (d) 6
  9. 9.How far is the point (12, 5) from the origin (0, 0)? (a) 17 (b) 169 (c) 14 (d) 13
  10. 10.A bag has 5 red balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 2 (c) 16 (d) 6
  11. 11.A cube has edges of 3 cm. What is its volume? (a) 36 cu cm (b) 27 cu cm (c) 54 cu cm (d) 9 cu cm
  12. 12.A bag has 2 red, 8 blue and 9 yellow balls. One ball is taken out without looking. What is the probability that it is blue? (a) 8/19 (b) 1/3 (c) 8/11 (d) 11/19
  13. 13.In a class of 36 children, 25 like maths, 20 like science and 11 like both. How many like at least one of the two? (a) 45 (b) 34 (c) 36 (d) 23
  14. 14.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  15. 15.Two angles of a triangle are 55° and 40°. What is the exterior angle at the third corner? (a) 125° (b) 140° (c) 85° (d) 95°
  16. 16.Two angles of a triangle are 70° and 90°. What is the third angle? (a) 90° (b) 110° (c) 20° (d) 160°
  17. 17.Three coins are tossed together. What is the probability of getting exactly one head? (a) 1/4 (b) 3/5 (c) 5/8 (d) 3/8
  18. 18.A book has 128 pages, numbered 1, 2, 3 and so on. How many digits are printed to number all the pages? (a) 276 (b) 277 (c) 267 (d) 256
  19. 19.What is the sum of all the whole numbers from 1 to 64? (a) 4160 (b) 2048 (c) 2080 (d) 2016
  20. 20.A drawer has lots of yellow and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 5 (c) 3 (d) 4
  21. 21.A bag has 8 blue, 9 red and 9 green balls. One ball is taken out without looking. What is the probability that it is blue? (a) 4/13 (b) 1/3 (c) 9/13 (d) 4/9
  22. 22.In how many different ways can the letters of the word CHAIR be arranged? (a) 120 (b) 25 (c) 20 (d) 240
  23. 23.Two coins are tossed together. What is the probability of getting a head on every coin? (a) 1/3 (b) 1/4 (c) 1/2 (d) 3/4
  24. 24.A bag has 6 black balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 8 (b) 7 (c) 9 (d) 2
  25. 25.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3364 (d) 3596
  26. 26.What is the median of 30, 40, 7, 26 and 10? (a) 7 (b) 30 (c) 10 (d) 26
  27. 27.In how many different ways can the letters of the word PLANT be arranged? (a) 25 (b) 20 (c) 240 (d) 120
  28. 28.A bag has 11 red balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 2 (c) 15 (d) 12
  29. 29.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  30. 30.A bag has 12 green balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 13 (b) 21 (c) 2 (d) 10
  31. 31.What is the mean of 7, 8, 14, 18 and 13? (a) 13 (b) 12 (c) 15 (d) 10
  32. 32.How many two-digit numbers have digits that add up to 5? (a) 5 (b) 10 (c) 4 (d) 6
  33. 33.How far is the point (16, 12) from the origin (0, 0)? (a) 21 (b) 400 (c) 20 (d) 28
  34. 34.In a class of 39 children, 15 like dance, 21 like music and 6 like both. How many like at least one of the two? (a) 36 (b) 24 (c) 30 (d) 39
  35. 35.A cube has edges of 6 cm. What is its volume? (a) 216 cu cm (b) 18 cu cm (c) 217 cu cm (d) 144 cu cm
  36. 36.What is the sum of the first 21 odd numbers (1 + 3 + 5 + …)? (a) 441 (b) 41 (c) 462 (d) 231
  37. 37.Two angles of a triangle are 65° and 105°. What is the third angle? (a) 75° (b) 115° (c) 10° (d) 170°
  38. 38.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 50 (c) 78 (d) 57
  39. 39.There are 6 roads from town A to town B and 4 roads from town B to town C. How many different routes go from A to C through B? (a) 30 (b) 10 (c) 48 (d) 24
  40. 40.What is the area of a circle of diameter 14 cm? (Use π = 22/7.) (a) 77 sq cm (b) 44 sq cm (c) 616 sq cm (d) 154 sq cm

Answer key

  1. 90
  2. 2² × 3 × 7
  3. 90
  4. 9
  5. 13
  6. 3/4
  7. 144
  8. 10
  9. 13
  10. 7
  11. 27 cu cm
  12. 8/19
  13. 34
  14. 157
  15. 95°
  16. 20°
  17. 3/8
  18. 276
  19. 2080
  20. 3
  21. 4/13
  22. 120
  23. 1/4
  24. 8
  25. 3364
  26. 26
  27. 120
  28. 12
  29. (x + 5)(x + 8)
  30. 13
  31. 12
  32. 5
  33. 20
  34. 30
  35. 216 cu cm
  36. 441
  37. 10°
  38. 50
  39. 24
  40. 154 sq cm

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