IMO Study Guide 2026
Everything you need to pass the IMO exam in one place: the exam format, every topic to study, real practice questions with explanations, flashcards, and full-length practice tests. Free, no sign-up needed.
📋 IMO Exam Format at a Glance
📚 IMO Topics to Study (23)
✍️ Sample IMO Questions & Answers
1. How many terms are in the arithmetic sequence: 7, 11, 15, ..., 55?
n = (last − first)/d + 1 = (55 − 7)/4 + 1 = 48/4 + 1 = 13.
2. 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.
Explanation: Let's denote the fourth number as x. Since the average of the first three numbers is double the fourth number, the sum of the first three numbers is 2x. Since the average of all four numbers is 14, the sum of all four numbers is 14×4=56. Thus, we have the equation 2x+x=56, which simplifies to 3x=56. Solving for x=56/3. The nearest whole number to 56/3 is 18.667, which rounds to 6 when rounded to the nearest whole number. Therefore, the fourth number is 8.
3. Which of the following is a composite number?
15 = 3 × 5, so it is composite because it has factors other than 1 and itself.
4. A jar has 4 green, 3 yellow, and 3 red candies. What is the probability of randomly picking a green candy?
Total candies = 10; P(green) = 4/10 = 2/5.
5. For positive reals x and y with x + y = 10, what is the maximum value of xy?
By AM-GM, xy ≤ ((x+y)/2)² = 25, with equality when x = y = 5.
6. What is the simple interest on $500 at 6% per year for 3 years?
Simple Interest = P × R × T / 100 = 500 × 6 × 3 / 100 = $90.