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.

1. How many different 3-digit numbers can be formed from 1, 2, 3, 5, 7 without repetition? → 60
2. In the Engel form (Titu's Lemma) of Cauchy-Schwarz, the inequality Σ(aᵢ²/bᵢ) ≥ (Σaᵢ)²/Σbᵢ is most directly useful for: → Establishing lower bounds on sums of fractions
3. What is the value of (−3) × (−5)? → 15
4. If C(n, 2) = 15, what is n? → 6
5. For positive reals a, b, c, what is the minimum value of (a+b)(b+c)(c+a) given abc = 1? → 8
6. What is the 7th term of the sequence 5, 10, 15, 20, ...? → 35
7. Solve for n: n/3 + 4 = 10 → 18
8. The Chebyshev sum inequality for similarly ordered sequences a₁ ≤ ··· ≤ aₙ and b₁ ≤ ··· ≤ bₙ states: → n · Σaᵢbᵢ ≥ (Σaᵢ)(Σbᵢ)
9. What is the product of the first 8 prime numbers? → 9,700
10. If a train covers 3/5 of a journey in 3 hours, how long will the full journey take? → 5 hours
11. Hölder's inequality is a generalization of which of the following classical inequalities? → Cauchy-Schwarz inequality
12. A triangle has sides of 5 cm, 5 cm, and 8 cm. What type of triangle is it? → Isosceles
13. What is the value of (48 + 32) ÷ (5 × 4)? → 4
14. In a circle, what is the longest chord called? → Diameter
15. What comes next in the pattern: 1, 4, 9, 16, 25, ___? → 36
16. What is 0.375 expressed as a fraction in its simplest form? → 3/8
17. 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
18. What is the sum of interior angles of a triangle? → 180°
19. For positive reals x, y, z with xyz = 8, what is the minimum of x + y + z? → 6
20. Which of the following is the correct order of operations for 3 + 4 × 2? → 11
21. Which of the following is a perfect square? → 64
22. A cube has a side length of 4 cm. What is its volume? → 64 cm³
23. 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
24. What is the LCM of 4 and 6? → 12
25. In a geometric sequence, the first term is 2 and the common ratio is 3. What is the 4th term? → 54
26. If 5n = 45, what is n? → 9
27. If the reflection of a clock in the mirror shows the time as 4:35, find the actual time. → 7:25
28. What is the area of a triangle with base 10 cm and height 6 cm? → 30 cm²
29. How many ways can a team of 3 be selected from 7 players? → 35
30. A jar has 4 green, 3 yellow, and 3 red candies. What is the probability of randomly picking a green candy? → 2/5