Problems

Find all positive integers \((x,n)\) such that \(x^n + 2^n + 1\) is a divisor \(x^{n+1} + 2^{n+1} + 1\).

Show that if \(n\) is an integer, greater than \(1\), then \(n\) does not divide \(2^n-1\).

Find all the integers \(n\) such that \(1^n + 2^n + ... + (n-1)^n\) is divisible by \(n\).

Take a pile of cards, Ace to 7 of Diamonds. Consider the following shuffle: simultaneously move the first card to the third position, the third card to the fifth position, the fifth card to the seventh position, and the seventh card to the first position. Also move the second card to the fourth position, the fourth card to the sixth position and the sixth card to the second position.

How many times do you have to do this one specific shuffle in a row to get back to where you started?

How many permutations are there of 4 cards leaving no card in the same position as before?

Lines \(AB\) and \(CDE\) are parallel. Which triangle out of \(\triangle ABC\), \(\triangle ABD\) and \(\triangle ABE\) has the greatest area?

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.