International Mathematical Olympiad (IMO) — Questions and Answers

Question 1: Solve for n: n/3 + 4 = 10

15
21
12
18 (Correct answer)

Correct answer: 18

n/3 = 10 − 4 = 6 → n = 6 × 3 = 18.

Question 2: Simplify: 3(2x − 4) + 5

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

Correct answer: 6x − 7

3(2x − 4) + 5 = 6x − 12 + 5 = 6x − 7.

Question 3: In a sequence 2, x, 18, 54, ..., what is the value of x if this is a geometric sequence?

8
6 (Correct answer)
9
4

Correct answer: 6

Common ratio r = 54/18 = 3; so x = 2 × 3 = 6.

Question 4: Which of the following is a perfect square?

64 (Correct answer)
90
72
50

Correct answer: 64

64 = 8², so it is a perfect square.

Question 5: If C(n, 2) = 15, what is n?

5
7
6 (Correct answer)
4

Correct answer: 6

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

Question 6: A coin is tossed 3 times. How many possible outcomes are there?

6
4
8 (Correct answer)
10

Correct answer: 8

Each toss has 2 outcomes; total = 2³ = 8.

Question 7: What is the value of 4³ − 3²?

63
43
55 (Correct answer)
52

Correct answer: 55

4³ − 3² = 64 − 9 = 55.

Question 8: 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 9: 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
Aunt
Nephew
Cousin (Correct answer)

Correct answer: Cousin

Explanation: <br> 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 10: 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.

Question 11: Which of the following is a composite number?

11
7
15 (Correct answer)
2

Correct answer: 15

15 = 3 × 5, so it is composite because it has factors other than 1 and itself.

Question 12: Two cards are drawn from a standard 52-card deck without replacement. How many ways can both cards be aces?

8
12
4
6 (Correct answer)

Correct answer: 6

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

Question 13: A menu has 4 starters and 3 main courses. How many different meals (one starter + one main) are possible?

10
12 (Correct answer)
7
14

Correct answer: 12

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

Question 14: How many ways can first, second, and third prizes be awarded to 6 contestants?

18
90
120 (Correct answer)
60

Correct answer: 120

P(6,3) = 6!/(6−3)! = 6 × 5 × 4 = 120.

Question 15: If p = 3 and q = 7, what is 2p + 3q?

20
27 (Correct answer)
35
29

Correct answer: 27

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

Question 16: A rectangular courtyard with a length of 3 m 25 cm and a breadth of 2 m 35 cm is to be paved with square stones of the same size. Find the least number of such stones required.

15275
560
3055 (Correct answer)
564

Correct answer: 3055

Explanation: <br> To find the least number of square stones required, divide the area of the rectangular courtyard by the area of one square stone. The dimensions of the courtyard are 325 cm by 235 cm, so its area is 76,375 square centimeters. Assuming each stone is x cm by x cm, we want to maximize x. The greatest common divisor (GCD) of 325 and 235 is 5, so each stone is 5 cm by 5 cm, covering 25 square centimeters. Dividing the area of the courtyard by the area of one stone gives us 3,055 stones.

Question 17: What is the value of 5a − 3 when a = 4?

20
23
11
17 (Correct answer)

Correct answer: 17

5a − 3 = 5 × 4 − 3 = 20 − 3 = 17.

Question 18: Which of the following is the correct order of operations for 3 + 4 × 2?

10
14
11 (Correct answer)
24

Correct answer: 11

By BODMAS/PEMDAS, multiplication comes before addition: 4 × 2 = 8, then 3 + 8 = 11.

Question 19: How many ways can a team of 3 be selected from 7 players?

42
70
35 (Correct answer)
21

Correct answer: 35

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

Question 20: The exterior angle of a regular polygon is 45°. How many sides does it have?

6
8 (Correct answer)
9
10

Correct answer: 8

Number of sides = 360° ÷ exterior angle = 360° ÷ 45° = 8.

Question 21: If 5n = 45, what is n?

9 (Correct answer)
8
10
7

Correct answer: 9

n = 45 ÷ 5 = 9.

Question 22: Which of these is NOT a factor of 48?

8
12
9 (Correct answer)
6

Correct answer: 9

48 ÷ 9 = 5.33…, which is not a whole number, so 9 is not a factor of 48.

Question 23: What is the name of a polygon with 6 sides?

Octagon
Pentagon
Hexagon (Correct answer)
Heptagon

Correct answer: Hexagon

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

Question 24: Two numbers have an HCF of 5 and an LCM of 60. If one number is 15, what is the other?

30
20 (Correct answer)
10
25

Correct answer: 20

Product of numbers = HCF × LCM = 5 × 60 = 300; other number = 300 ÷ 15 = 20.

Question 25: How many different 3-digit numbers can be formed from 1, 2, 3, 5, 7 without repetition?

50
40
30
60 (Correct answer)

Correct answer: 60

P(5,3) = 5 × 4 × 3 = 60.

Question 26: What is the value of 2⁵?

10
16
64
32 (Correct answer)

Correct answer: 32

2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Question 27: A tank is 3/4 full. If 90 liters are added to fill it completely, what is the tank's full capacity?

400 liters
270 liters
300 liters
360 liters (Correct answer)

Correct answer: 360 liters

90 liters represents 1/4 of the tank; full capacity = 90 × 4 = 360 liters.

Question 28: Which of the following is a linear equation?

y = x²
y = 3x + 2 (Correct answer)
y = x³ + 1
y = 1/x

Correct answer: y = 3x + 2

y = 3x + 2 is a linear equation because the highest power of x is 1.

Question 29: A shopkeeper bought goods for $450 and sold them for $540. What is the profit percentage?

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

Correct answer: 20%

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

Question 30: A lighthouse has two lights - one that flashes every 4 minutes and another that flashes every 3½ minutes. Suppose the lights flash together at noon. What is the first time after 2 PM when they will flash together again?

2:15
2:05
2:00
2:20 (Correct answer)

Correct answer: 2:20

Explanation: <br> The least common multiple (LCM) of 4 minutes and 3½ minutes is 7 minutes. Adding 7 minutes to 2:00 PM gives us 2:07 PM. Since 2:07 PM is not an option, the next time after 2:07 PM is 2:14 PM, but since the light flashes together at noon, the next time they flash together is 2:20 PM.

Question 31: Simplify: 7m + 3m − 2m

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

Correct answer: 8m

7m + 3m − 2m = 10m − 2m = 8m.

Question 32: What is the value of 4k² when k = 2?

16 (Correct answer)
32
64
8

Correct answer: 16

4k² = 4 × 2² = 4 × 4 = 16.

Question 33: A die is rolled once. What is the probability of getting an even number?

2/3
1/6
1/2 (Correct answer)
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 34: In how many ways can a committee of 2 be chosen from 5 people?

10 (Correct answer)
5
20
8

Correct answer: 10

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

Question 35: What is the sum of all prime numbers less than 10?

19
17 (Correct answer)
15
23

Correct answer: 17

Primes less than 10 are 2, 3, 5, 7; their sum = 2 + 3 + 5 + 7 = 17.

Question 36: What are the next two numbers in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, _, _?

11, 18
12, 20
13, 21 (Correct answer)
14, 22

Correct answer: 13, 21

Each term is the sum of the two preceding terms: 8 + 5 = 13, then 13 + 8 = 21.

Question 37: In how many ways can 4 people be seated in a row of 4 chairs?

24 (Correct answer)
12
16
20

Correct answer: 24

This is 4! = 4 × 3 × 2 × 1 = 24.

Question 38: If 3x + 7 = 22, what is the value of x?

5 (Correct answer)
6
3
4

Correct answer: 5

3x = 22 − 7 = 15, so x = 15 ÷ 3 = 5.

Question 39: 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 40: For positive reals x and y with x + y = 10, what is the maximum value of xy?

100
20
25 (Correct answer)
10

Correct answer: 25

By AM-GM, xy ≤ ((x+y)/2)² = 25, with equality when x = y = 5.

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%

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