Practice Test GeeksIMO - International Mathematics Olympiad Practice Test

IMO Cheat Sheet 2026

The 30 highest-yield IMO facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

IMO exam key facts

  • The International Mathematical Olympiad has 6 problems, each worth 7 points, for a maximum total score of 42. Source
  • The IMO has no passing score; the number of medals must not exceed half of all contestants, gold, silver and bronze go in roughly a 1:2:3 ratio, and contestants without a medal who fully solve one problem receive an Honourable Mention. Source
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40 questions
120 min time limit
70.00% to pass
  1. How many different 3-digit numbers can be formed from 1, 2, 3, 5, 7 without repetition? → 60
  2. What is the value of (−3) × (−5)? → 15
  3. If C(n, 2) = 15, what is n? → 6
  4. What is the 7th term of the sequence 5, 10, 15, 20, ...? → 35
  5. Solve for n: n/3 + 4 = 10 → 18
  6. What is the product of the first 8 prime numbers? → 9,700
  7. If a train covers 3/5 of a journey in 3 hours, how long will the full journey take? → 5 hours
  8. A triangle has sides of 5 cm, 5 cm, and 8 cm. What type of triangle is it? → Isosceles
  9. What is the value of (48 + 32) ÷ (5 × 4)? → 4
  10. In a circle, what is the longest chord called? → Diameter
  11. What comes next in the pattern: 1, 4, 9, 16, 25, ___? → 36
  12. What is 0.375 expressed as a fraction in its simplest form? → 3/8
  13. Tap A can fill a tank in 4 hours, while tap B can empty the full tank in 5 hours. If both taps are open together, how long will it take to fill an empty tank? → 20 hours
  14. What is the sum of interior angles of a triangle? → 180°
  15. Which of the following is the correct order of operations for 3 + 4 × 2? → 11
  16. Which of the following is a perfect square? → 64
  17. A cube has a side length of 4 cm. What is its volume? → 64 cm³
  18. The average of four numbers is 14. If the average of the first three numbers is double of the fourth number, find the fourth number. → 8
  19. What is the LCM of 4 and 6? → 12
  20. In a geometric sequence, the first term is 2 and the common ratio is 3. What is the 4th term? → 54
  21. If 5n = 45, what is n? → 9
  22. If the reflection of a clock in the mirror shows the time as 4:35, find the actual time. → 7:25
  23. What is the area of a triangle with base 10 cm and height 6 cm? → 30 cm²
  24. How many ways can a team of 3 be selected from 7 players? → 35
  25. A jar has 4 green, 3 yellow, and 3 red candies. What is the probability of randomly picking a green candy? → 2/5
  26. The sum of two numbers is 50. One number is 18. What is the other? → 32
  27. In a sequence 2, x, 18, 54, ..., what is the value of x if this is a geometric sequence? → 6
  28. In how many ways can you arrange 3 books on a shelf? → 6
  29. Which of the following is a composite number? → 15
  30. A student scored 72 out of 90. What percentage did they score? → 80%
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How this page was made: written with AI tools and published by Nathan Beja, founder of Practice Test Geeks, who is responsible for its content. Exam facts are checked against the official sources listed below. Not reviewed by a subject-matter expert. Editorial policy

Sources & references

Exam facts on this page (fees, format, dates, passing scores) are checked against these official sources; the date is our last check. Editorial policy

  1. imo-official.org — Test delivery, eligibility, passing score, test dates, exam format · verified Oct 2026
  2. imo2027.hu — Exam dates, registration deadline · verified Oct 2026