International Mathematical Olympiad (IMO) — Questions and Answers

Question 1: In how many ways can a committee of 2 be chosen from 5 people?

5
20
10 (Correct answer)
8

Correct answer: 10

C(5,2) = 5!/(2! × 3!) = (5 × 4)/(2 × 1) = 10.

Question 2: While doing a science experiment in the physics lab, Pranav took 5 measurements of the temperature and wrote the average of those as an answer. If the measurements of the temperature are -3, 2, 1, -1 and -3 then what is the final answer of his experiment?

0.2
1.2
2.2
-0.8 (Correct answer)

Correct answer: -0.8

Explanation: To find the average temperature, we add up all the measurements and then divide by the total number of measurements. Sum of measurements: (−3+2+1−1−3)=−4 Total number of measurements: 5 Average temperature: (−4)/5=−0.8

Question 3: What is the value of 5a − 3 when a = 4?

11
17 (Correct answer)
20
23

Correct answer: 17

5a − 3 = 5 × 4 − 3 = 20 − 3 = 17.

Question 4: A number leaves a remainder of 3 when divided by 7. Which could be the number?

24 (Correct answer)
31
38
17

Correct answer: 24

24 ÷ 7 = 3 remainder 3, so 24 leaves a remainder of 3 when divided by 7.

Question 5: What is the HCF (GCD) of 18 and 24?

9
4
6 (Correct answer)
3

Correct answer: 6

The Highest Common Factor of 18 and 24 is 6, as 6 is the largest number that divides both.

Question 6: What is the probability of getting a sum of 7 when two dice are rolled?

1/6
5/36
6/36 (Correct answer)
7/36

Correct answer: 6/36

Combinations that give 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) — 6 out of 36 total outcomes = 6/36 = 1/6.

Question 7: Simplify: 3(2x − 4) + 5

6x − 7 (Correct answer)
6x − 1
5x − 7
6x + 5

Correct answer: 6x − 7

3(2x − 4) + 5 = 6x − 12 + 5 = 6x − 7.

Question 8: What is the result of 5/8 + 3/8?

15/64
7/8
1 (Correct answer)
8/16

Correct answer: 1

5/8 + 3/8 = 8/8 = 1.

Question 9: Which pattern shows square numbers?

2, 6, 12, 20, 30
2, 4, 6, 8, 10
1, 3, 6, 10, 15
1, 4, 9, 16, 25 (Correct answer)

Correct answer: 1, 4, 9, 16, 25

1, 4, 9, 16, 25 are the squares of 1, 2, 3, 4, and 5 respectively.

Question 10: The Cauchy-Schwarz inequality for real sequences a₁, …, aₙ and b₁, …, bₙ states:

(Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²) (Correct answer)
Σaᵢbᵢ = √(Σaᵢ² · Σbᵢ²)
Σaᵢbᵢ ≤ n·max(aᵢbᵢ)
(Σaᵢbᵢ)² ≥ (Σaᵢ²)(Σbᵢ²)

Correct answer: (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)

The Cauchy-Schwarz inequality states (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²), with equality when the sequences are proportional.

Question 11: If p = 3 and q = 7, what is 2p + 3q?

20
35
27 (Correct answer)
29

Correct answer: 27

2p + 3q = 2(3) + 3(7) = 6 + 21 = 27.

Question 12: How many 2-digit numbers can be formed using digits 1, 2, 3, 4 (without repetition)?

8
16
12 (Correct answer)
10

Correct answer: 12

There are 4 choices for the first digit and 3 for the second: 4 × 3 = 12.

Question 13: If 3x + 7 = 22, what is the value of x?

3
5 (Correct answer)
4
6

Correct answer: 5

3x = 22 − 7 = 15, so x = 15 ÷ 3 = 5.

Question 14: Which expression is equivalent to 2(x + 5)?

2x + 10 (Correct answer)
2x + 5
2x + 7
x + 10

Correct answer: 2x + 10

Using the distributive property: 2(x + 5) = 2x + 10.

Question 15: Showing the man singing on the stage, Sarika says, "He is the brother of my uncle's daughter". How is Sarika related to the man?

Daughter
Nephew
Aunt
Cousin (Correct answer)

Correct answer: Cousin

Explanation: Sarika's uncle's daughter is her cousin. The man singing on the stage is stated to be the brother of Sarika's cousin, so he is also Sarika's cousin.

Question 16: 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?

21 hours
19 hours
23 hours
20 hours (Correct answer)

Correct answer: 20 hours

Explanation: Tap A fills the tank in 4 hours, so in 1 hour it fills 1/4 of the tank. Tap B empties the tank in 5 hours, so in 1 hour it empties 1/5 of the tank. When both taps are open together, they work simultaneously, so the net filling rate is (1/4)−(1/5)=1/20 of the tank per hour. Therefore, it takes 20 hours to fill the tank.

Question 17: A bag has 5 red and 3 blue balls. What is the probability of picking a blue ball?

3/8 (Correct answer)
1/3
3/5
5/8

Correct answer: 3/8

Total balls = 8; P(blue) = 3/8.

Question 18: Which value of x satisfies 3x > 12?

x = 2
x = 5 (Correct answer)
x = 3
x = 4

Correct answer: x = 5

3x > 12 → x > 4, so x = 5 is the only option greater than 4.

Question 19: Using the letters A, B, C, D, E, how many 3-letter arrangements can be made without repetition?

120
60 (Correct answer)
30
10

Correct answer: 60

P(5,3) = 5!/(5−3)! = 5 × 4 × 3 = 60.

Question 20: What is the value of (48 + 32) ÷ (5 × 4)?

6
4 (Correct answer)
2
8

Correct answer: 4

(48 + 32) ÷ (5 × 4) = 80 ÷ 20 = 4.

Question 21: Divya plays squash every 3rd day and plays football every 4th day. She played both squash and football on 4th April. What is the next date when Divya will play both games?

17th April
15th April
14th April
16th April (Correct answer)

Correct answer: 16th April

Explanation: Divya plays squash every 3rd day and football every 4th day. The next date when she plays both games will be the least common multiple (LCM) of 3 and 4, which is 12. Adding 12 days to April 4th, we get April 16th, when Divya will play both squash and football again.

Question 22: What is 15% of 200?

25
35
30 (Correct answer)
20

Correct answer: 30

15% of 200 = (15/100) × 200 = 30.

Question 23: A die is rolled once. What is the probability of getting an even number?

1/2 (Correct answer)
2/3
1/6
1/3

Correct answer: 1/2

Even numbers on a die: 2, 4, 6 — that's 3 out of 6; probability = 3/6 = 1/2.

Question 24: What is the product of (x + 2)(x + 3)?

x² + 6x + 6
x² + 5x + 5
x² + 5x + 6 (Correct answer)
x² + 6x + 5

Correct answer: x² + 5x + 6

(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.

Question 25: What are the prime factors of 36?

2, 3, and 6
3 and 4
4 and 9
2 and 3 (Correct answer)

Correct answer: 2 and 3

36 = 2² × 3², so its prime factors are 2 and 3.

Question 26: A shopkeeper bought goods for $450 and sold them for $540. What is the profit percentage?

15%
20% (Correct answer)
18%
25%

Correct answer: 20%

Profit = $90; Profit % = (90/450) × 100 = 20%.

Question 27: How many ways can a team of 3 be selected from 7 players?

21
70
42
35 (Correct answer)

Correct answer: 35

C(7,3) = 7!/(3! × 4!) = (7 × 6 × 5)/(3 × 2 × 1) = 35.

Question 28: A rectangle has a length of 12 cm and a width of 5 cm. What is its perimeter?

34 cm (Correct answer)
24 cm
17 cm
60 cm

Correct answer: 34 cm

Perimeter = 2 × (length + width) = 2 × (12 + 5) = 2 × 17 = 34 cm.

International Mathematical Olympiad (IMO)

The International Mathematical Olympiad (IMO) is the world's most prestigious mathematics competition for high school students, held annually since 1959. Competitors solve six proof-based problems over two days covering algebra, number theory, combinatorics, and geometry, with each problem worth up to 7 points for a maximum score of 42.

Questions

35 min

Time Limit

70%

To Pass

Checking for saved progress...

Exam Rules

You can skip questions and return to them later
Flag questions for review before submitting
No feedback shown until you submit the entire exam
Unanswered questions count as wrong — answer everything
10 pretest questions are mixed in and don't affect your score
Timer auto-submits when time runs out
Your progress is auto-saved every 30 seconds

These practice questions were written with AI tools. They are not real exam questions. 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

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