International Mathematical Olympiad (IMO) — Questions and Answers

Question 1: Which of the following is the greatest: 3/4, 5/8, 7/10, or 2/3?

2/3
3/4 (Correct answer)
5/8
7/10

Correct answer: 3/4

Converting to decimals: 3/4 = 0.75, 5/8 = 0.625, 7/10 = 0.70, 2/3 ≈ 0.667; so 3/4 is greatest.

Question 2: In how many ways can 4 people be seated in a row of 4 chairs?

12
16
20
24 (Correct answer)

Correct answer: 24

This is 4! = 4 × 3 × 2 × 1 = 24.

Question 3: There are 60 questions in a competitive exam. 3 marks are awarded for each correct answer, while 2 marks are deducted for each incorrect answer. Archana answered 47 questions correctly. If she attempts all the questions, how many marks did she get?

115 (Correct answer)
55
123
127

Correct answer: 115

Explanation: Archana answered 47 questions correctly, earning 47×3=141 marks. She attempted all 60 questions, so the number of incorrect answers is 60−47=13. For each incorrect answer, 2 marks are deducted, totaling 13×2=26 marks. Subtracting the deducted marks from the earned marks, Archana's total marks are 141−26=115. So, she got 115 marks.

Question 4: In how many ways can a committee of 2 be chosen from 5 people?

8
10 (Correct answer)
20
5

Correct answer: 10

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

Question 5: What is the name of a polygon with 6 sides?

Pentagon
Octagon
Heptagon
Hexagon (Correct answer)

Correct answer: Hexagon

A hexagon is a polygon with exactly six sides and six angles.

Question 6: Solve: 4x − 6 = 2x + 8

9
5
11
7 (Correct answer)

Correct answer: 7

4x − 2x = 8 + 6 → 2x = 14 → x = 7.

Question 7: A recipe needs 2.5 cups of flour for 10 cookies. How many cups are needed for 24 cookies?

5 cups
6 cups (Correct answer)
7 cups
8 cups

Correct answer: 6 cups

Rate = 2.5/10 = 0.25 cups per cookie; 24 × 0.25 = 6 cups.

Question 8: How many 2-digit numbers can be formed using digits 1, 2, 3, 4 (without repetition)?

12 (Correct answer)
10
8
16

Correct answer: 12

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

Question 9: What is the simple interest on $500 at 6% per year for 3 years?

$72
$100
$90 (Correct answer)
$80

Correct answer: $90

Simple Interest = P × R × T / 100 = 500 × 6 × 3 / 100 = $90.

Question 10: 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 11: What is C(6, 2)?

30
15 (Correct answer)
18
12

Correct answer: 15

C(6,2) = 6!/(2! × 4!) = (6 × 5)/(2 × 1) = 15.

Question 12: Solve for n: n/3 + 4 = 10

21
15
18 (Correct answer)
12

Correct answer: 18

n/3 = 10 − 4 = 6 → n = 6 × 3 = 18.

Question 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?

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

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 14: In how many ways can you arrange 3 books on a shelf?

9
3
6 (Correct answer)
12

Correct answer: 6

The number of arrangements = 3! = 3 × 2 × 1 = 6.

Question 15: Which of the following is a perfect square?

64 (Correct answer)
90
50
72

Correct answer: 64

64 = 8², so it is a perfect square.

Question 16: What is the probability of getting a sum of 7 when two dice are rolled?

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

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 17: What is the value of 4k² when k = 2?

32
16 (Correct answer)
64
8

Correct answer: 16

4k² = 4 × 2² = 4 × 4 = 16.

Question 18: A right-angled triangle has legs of length 3 cm and 4 cm. What is the length of the hypotenuse?

7 cm
5 cm (Correct answer)
6 cm
8 cm

Correct answer: 5 cm

By the Pythagorean theorem: hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.

Question 19: Simplify: 3(2x − 4) + 5

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

Correct answer: 6x − 7

3(2x − 4) + 5 = 6x − 12 + 5 = 6x − 7.

Question 20: While doing a science experiment in the physics lab, Sanjana took 7 measurements of the temperature and wrote the average of those as an answer. If the measurements of the temperature are -2, 1, 2, 0, -4, 2 and 2 then what is the final answer of her experiment?

3.14
2.14
1.14
0.14 (Correct answer)

Correct answer: 0.14

Explanation: To find the average temperature, we add up all the measurements and then divide by the total number of measurements. Sum of measurements: −2+1+2+0−4+2+2=1 Total number of measurements: 7 Average temperature: 1/7=0.14 So, the final answer to her experiment is 0.14.

Question 21: Which of the following numbers is a prime number?

21
29 (Correct answer)
27
33

Correct answer: 29

29 is prime because it has no divisors other than 1 and itself.

Question 22: If C(n, 2) = 15, what is n?

6 (Correct answer)
7
5
4

Correct answer: 6

C(n,2) = n(n−1)/2 = 15 → n(n−1) = 30 → n = 6 (since 6 × 5 = 30).

Question 23: If 3x + 7 = 22, what is the value of x?

5 (Correct answer)
3
4
6

Correct answer: 5

3x = 22 − 7 = 15, so x = 15 ÷ 3 = 5.

Question 24: For a convex function f and real numbers x, y, Jensen's inequality gives:

f((x+y)/2) ≥ (f(x)+f(y))/2
f(x+y) = f(x)+f(y)
f(x+y) ≤ f(x)·f(y)
f((x+y)/2) ≤ (f(x)+f(y))/2 (Correct answer)

Correct answer: f((x+y)/2) ≤ (f(x)+f(y))/2

For a convex function, Jensen's inequality states f((x+y)/2) ≤ (f(x)+f(y))/2, reflecting that the function lies below the chord between two points.

Question 25: Which of the following numbers is divisible by both 3 and 4?

30
24 (Correct answer)
18
20

Correct answer: 24

24 ÷ 3 = 8 and 24 ÷ 4 = 6, so 24 is divisible by both 3 and 4.

Question 26: What is 3⁴?

27
81 (Correct answer)
64
12

Correct answer: 81

3⁴ = 3 × 3 × 3 × 3 = 81.

Question 27: Two cards are drawn from a standard 52-card deck without replacement. How many ways can both cards be aces?

6 (Correct answer)
8
4
12

Correct answer: 6

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

Question 28: 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 (Correct answer)
Computing exact values of symmetric sums
Proving products of sums are bounded
Deriving upper bounds on geometric means

Correct answer: Establishing lower bounds on sums of fractions

Titu's Lemma (Engel form) is used to establish lower bounds on sums of the form Σ(aᵢ²/bᵢ) by combining the fractions into a single expression.

Question 29: Using the letters A, B, C, D, E, how many 3-letter arrangements can be made without repetition?

60 (Correct answer)
10
30
120

Correct answer: 60

P(5,3) = 5!/(5−3)! = 5 × 4 × 3 = 60.

Question 30: What is the HCF (GCD) of 18 and 24?

4
3
6 (Correct answer)
9

Correct answer: 6

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

Question 31: Which expression is equivalent to 2(x + 5)?

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

Correct answer: 2x + 10

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

Question 32: A menu has 4 starters and 3 main courses. How many different meals (one starter + one main) are possible?

10
14
7
12 (Correct answer)

Correct answer: 12

Total combinations = 4 × 3 = 12 by the multiplication principle.

Question 33: Simplify: 7m + 3m − 2m

6m
8m (Correct answer)
7m
12m

Correct answer: 8m

7m + 3m − 2m = 10m − 2m = 8m.

Question 34: What is the missing term: 100, 81, 64, ___, 36?

50
48
49 (Correct answer)
54

Correct answer: 49

These are perfect squares in decreasing order: 10², 9², 8², 7², 6²; the missing term is 7² = 49.

Question 35: What is the product of (x + 2)(x + 3)?

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

Correct answer: x² + 5x + 6

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

Question 36: Determine the two numbers nearest to 9000 which are exactly divisible by 4, 9, 6, and 5.

9000, 9182
9001, 9180
8999, 9180
9000, 9180 (Correct answer)

Correct answer: 9000, 9180

Explanation: To find numbers divisible by 4, 9, 6, and 5 nearest to 9000, we need to find the least common multiple (LCM) of these numbers, which is 180. The nearest numbers divisible by 180 around 9000 are 9000 and 9180.

Question 37: If 2(x + 3) = 18, what is x?

6 (Correct answer)
7
9
5

Correct answer: 6

2(x + 3) = 18 → x + 3 = 9 → x = 6.

Question 38: What is the value of (−3) × (−5)?

−15
−8
8
15 (Correct answer)

Correct answer: 15

The product of two negative numbers is positive: (−3) × (−5) = 15.

Question 39: What is the value of √144?

14
12 (Correct answer)
11
13

Correct answer: 12

√144 = 12 because 12² = 144.

Question 40: Which classical inequality states that for positive reals a and b, (a + b) / 2 ≥ √(ab)?

AM-GM inequality (Correct answer)
Cauchy-Schwarz inequality
Chebyshev's inequality
Jensen's inequality

Correct answer: AM-GM inequality

The AM-GM inequality (Arithmetic Mean – Geometric Mean) states that the arithmetic mean of non-negative reals is always at least their geometric mean.

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

120 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