Problems

In a square, the midpoints of its sides were marked and connected to the vertices of the square. There is another square formed in the centre. Find its area, if the side of the square has length \(10\).

In a regular hexagon, some diagonals were drawn. Find the area of the red region, if the total area of the hexagon is \(72\).

Three semicircles are drawn on the sides of the triangle \(ABC\) with sides \(AB=3\), \(AC=4\), \(BC=5\) as diameters. Find the area of the red part.

\(20\) birds fly into a photographer’s studio: \(8\) starlings, \(7\) wagtails and \(5\) woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and does not come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than \(5\) birds of one species and no less than 3 of another species remaining in the studio.

Prove that among \(11\) different infinite decimal fractions, you can choose two fractions which coincide in an infinite number of digits.

A convex polygon on the plane contains at least \(m^2+1\) points with integer coordinates. Prove that it contains \(m+1\) points with integers coordinates that lie on the same line.

Suppose a football team scores at least one goal in each of the \(20\) consecutive games. If it scores a total of \(30\) goals in those \(20\) games, prove that in some sequence of consecutive games it scores exactly \(9\) goals total.

Find all the prime numbers \(p\) such that the number \(2p^2+1\) is also prime.

Show that a bipartite graph with \(n\) vertices cannot have more than \(\frac{n^2}{4}\) edges.

Now there are finitely many infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. Now what do you do?