4

Part of 2012 Irish Math Olympiad

Problems(2)

Triangles with Integer Sides

Source: 2012 IrMO Paper 1 Problem 4

There exists an infinite set of triangles with the following properties:
(a) the lengths of the sides are integers with no common factors, and
(b) one and only one angle is

60^\circ

.
One such triangle has side lengths

5,7,8

. Find two more.

Prime number question

Source:

x

1

be an integer. Prove that

x^5

x

1

is divisible by at least two distinct prime numbers.

number theoryprime numbers