Featured Questions on Mathematics Olympiad (IMO) For Classes 9 and 10

Mathematics Olympiad (IMO) questions are designed to test deep mathematical understanding and problem-solving skills. Here are some sample questions that reflect the style and complexity of IMO problems:

Mathematics Olympiad Questions for Class 10

1. Algebra

1.1 Problem 1:
Solve the quadratic equation:
2x^2 - 5x + 3 = 0

1.2 Problem 2:
Find the value of k for which the quadratic equation x^2 + (k-1)x + 2k = 0 has equal roots.

1.3 Problem 3:
Solve the equation for x:
3x^2 - 7x + 2 = 0

1.4 Problem 4:
Solve for x in the equation:
(2x + 3) / (x - 1) = 4

1.5 Problem 5:
If x and y are the roots of the quadratic equation x^2 - 5x + 6 = 0, find the value of x^3 + y^3.

1.6 Problem 6:
Determine the sum and product of the roots of the equation:
4x^2 + 7x - 3 = 0

1.7 Problem 7:
Find the roots of the quadratic equation:
x^2 - 6x + 8 = 0

1.8 Problem 8:
Solve the equation:
x^2 + 4x + 4 = 0

2. Geometry

2.1 Problem 1:
In triangle ABC, if ∠A = 60°, AB = AC, and BC = 10 units, find the lengths of AB and AC.

2.2 Problem 2:
In a right-angled triangle, where the hypotenuse is 13 units and one leg is 5 units, find the length of the other leg.

2.3 Problem 3:
Find the area of a triangle with base 8 units and height 5 units.

2.4 Problem 4:
In a circle with radius 7 units, find the length of the chord that is 5 units away from the center of the circle.

2.5 Problem 5:
Calculate the circumference of a circle with a diameter of 10 units.

2.6 Problem 6:
In a rectangle, if the length is twice the width and the perimeter is 60 units, find the dimensions of the rectangle.

2.7 Problem 7:
Find the area of a right-angled triangle where the legs are 9 units and 12 units long.

2.8 Problem 8:
In a parallelogram, if the base is 8 units and the height is 5 units, find the area of the parallelogram.

3. Number Theory

3.1 Problem 1:
Find the greatest common divisor (GCD) of 56 and 98.

3.2 Problem 2:
Find the sum of all positive integer solutions of the equation:
x^2 - 4y^2 = 1

3.3 Problem 3:
Determine the least common multiple (LCM) of 12 and 18.

3.4 Problem 4:
Find the number of positive divisors of 84.

3.5 Problem 5:
If a and b are two positive integers such that GCD(a, b) = 1 and LCM(a, b) = 60, find the possible pairs of (a, b).

3.6 Problem 6:
Find the remainder when 123456 is divided by 7.

3.7 Problem 7:
If p is a prime number and 2p - 1 is also a prime number, find the value of p.

3.8 Problem 8:
Determine whether 37 is a prime number and justify your answer.

4. Combinatorics

4.1 Problem 1:
How many ways can you arrange the letters in the word "MATHEMATICS"?

4.2 Problem 2:
In how many ways can 5 people be seated in a row?

4.3 Problem 3:
How many different 4-digit numbers can be formed using the digits 0, 1, 2, 3, and 4 if repetition is allowed?

4.4 Problem 4:
From a group of 10 people, how many ways can you select a committee of 3 people?

4.5 Problem 5:
In how many ways can you distribute 10 identical candies among 4 children?

4.6 Problem 6:
Calculate the number of ways to arrange the letters in the word "BANANA".

4.7 Problem 7:
How many ways can you choose 2 cards from a standard deck of 52 cards?

4.8 Problem 8:
If a password consists of 3 letters followed by 2 digits, how many possible passwords can be created if repetition is allowed?

Answers

1.1 Problem 1:
The solutions are:
x = 1 and x = 1.5

1.2 Problem 2:
The value of k is:
k = 5 ± 2√6

1.3 Problem 3:
The roots are:
x = 2 and x = 1/3

1.4 Problem 4:
The solution is:
x = 2.5

1.5 Problem 5:
The value of x^3 + y^3 is:
63

1.6 Problem 6:
The sum is:
-7 and the product is:
-3

1.7 Problem 7:
The roots are:
x = 2 and x = 4

1.8 Problem 8:
The root is:
x = -2

2.1 Problem 1:
The lengths of AB and AC are:
10 units

2.2 Problem 2:
The length of the other leg is:
12 units

2.3 Problem 3:
The area is:
20 square units

2.4 Problem 4:
The length of the chord is:
10 units

2.5 Problem 5:
The circumference is:
31.4 units

2.6 Problem 6:
The dimensions are:
Length = 20 units and Width = 10 units

2.7 Problem 7:
The area is:
54 square units

2.8 Problem 8:
The area is:
40 square units

3.1 Problem 1:
The GCD is:
14

3.2 Problem 2:
The sum of all positive integer solutions is:
6

3.3 Problem 3:
The LCM is:
36

3.4 Problem 4:
The number of positive divisors is:
12

3.5 Problem 5:
The possible pairs are:
(1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10)

3.6 Problem 6:
The remainder is:
2

3.7 Problem 7:
The value of p is:
2

3.8 Problem 8:
Yes, 37 is a prime number because it has no positive divisors other than 1 and itself.

4.1 Problem 1:
The number of distinct arrangements is:
831600

4.2 Problem 2:
The number of ways is:
120

4.3 Problem 3:
The number of different 4-digit numbers is:
625

4.4 Problem 4:
The number of ways is:
120

4.5 Problem 5:
The number of ways is:
286

4.6 Problem 6:
The number of distinct arrangements is:
60

4.7 Problem 7:
The number of ways to choose 2 cards is:
1326

4.8 Problem 8:
The number of possible passwords is:
17576000