Problems

In the picture below, we have a regular pentagon. The segments \(AB\) and \(CD\) have equal lengths. What is the angle \(\alpha\)?

We have two squares sharing the same centre, each with side length \(2\). Show that the area of overlap is at least \(3\).

A regular tetrahedron is a three dimensional shape with four faces. Each face of a regular tetrahedron is an equilateral triangle. Describe all rotational symmetries of a regular tetrahedron.

Two lines intersect at a point \(P\) at an angle of \(\alpha\). Show that a rotation in the plane around the point \(P\) through an angle \(2\alpha\) can be achieved by a reflection in one of the two lines followed by a reflection in the other line.

Let \(u\) and \(v\) be two positive integers, with \(u>v\). Prove that a triangle with side lengths \(u^2-v^2\), \(2uv\) and \(u^2+v^2\) is right-angled.

We call a triple of natural numbers (also known as positive integers) \((a,b,c)\) satisfying \(a^2+b^2=c^2\) a Pythagorean triple. If, further, \(a\), \(b\) and \(c\) are relatively prime, then we say that \((a,b,c)\) is a primitive Pythagorean triple.

Show that every primitive Pythagorean triple can be written in the form \((u^2-v^2,2uv,u^2+v^2)\) for some coprime positive integers \(u>v\).

What symmetries does a regular hexagon have, and how many?

The lengths of three sides of a right-angled triangle are all integers.

Show that one of them is divisible by \(5\).

Given a pile of five cards, is it true that reversing the order of the pile by counting the cards out one by one leaves no card in its original position?

With a pile of four cards, does reversing the order of the pile by counting the cards out one by one leaves no card in its original position?