Problem 31

A two-digit integer n has the property that the sum of n and its digits is equal to 63. Find n.

Problem 30

Compare 65²³ and 255¹⁷.

Problem 29

Factorise P(x) =  2x⁷ + 5x⁴ + 3.

Problem 28

Let ABC an isoceles triangle with AB = AC = x, BC = y. Let denote M the midpoint of AB. Find the length of CM.

Problem 27

A triangle has side lengths 15, 13 and 4 cm. Find the length of the longest altitude.

Problem 26

Prove that every composite number has a proper factor less than or equal to its square root.

Problem 25

Show that 43ⁿ + 83 x 92^3n-1 is divisible by 7 for all positive integers n.

Problem 24

In a hall, there are 1098 seats. There are two aisles. Each row has the same number of seats. In each row, the number of seats between the aisles is 31. The rest of the seats are equally divided between the aisles. Find the number of rows.

Problem 23

If f: R→ R (where R is the set of all real numbers) is given by f(f(x)) = 4x + 6. Find f(x).

Problem 22

If 22 January 2017 is a Sunday, what days was 30 April 1975?

Problem 21

Prove that the a composite n non divisible by   is a semiprime.

Problem 20

Find the last digit of 43⁴³.

Problem 19

Find the last digit of 2⁹⁹⁹.

Problem 18

Given a Pythagorean triple a, b, c, prove that if a is odd, then either b or c must be odd.

Problem 17

Prove that for any m, n,  m² - n², 2mn,  and m² + n² form a Pythagorean triple.

Problem 16

Prove that there are an infinite number of Pythagorean triples.

Problem 15

Prove that if n is an odd number, then n, 0.5 (n² - 1) and 0.5 (n² + 1) form a Pythagorean triple.

Problem 14

Fill in the missing digit to make the number 66* divisible by 17.

Problem 13

Prove that the square root of 201720172017  is an irrational number.

Problem 12

Prove that for all natural numbers n, the formula

 is true.

Problem 11

Prove that for all natural numbers n, the formula

 is true.