Problems

There are \(16\) cities in the kingdom. Prove that it is not possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most four roads coming out of each city.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a square, or slightly harder in a shape of a given rectangle.

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a shape like this

Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a given triangle.

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a circle.

The marked angles are all \(45^{\circ}\). Show that the total green and blue areas are the same.

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(AB\), point \(F\) belongs to the side \(CD\) and point \(G\) belongs to the side \(AD\). We know that the marked red segments \(AE\) and \(CF\) have equal lengths. Prove that the total grey area is equal to the total black area.

In every group of \(9\) randomly chosen crayons from Jamie’s drawer, some \(3\) will have the same colour. Show that if Jamie chooses \(25\) crayons at random, some \(7\) will have the same colour.

Suppose \(n \ge 2\) cricket teams play in a tournament. If no two teams play each other more than once, prove that some two teams have to play the same number of games.

Find all the prime numbers \(p\) such that there exist natural numbers \(x\) and \(y\) for which \(p^x = y^3 + 1\).