10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.
Every point in a plane, which has whole-number co-ordinates, is plotted in one of \(n\) colours. Prove that there will be a rectangle made out of 4 points of the same colour.
On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.
Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.
A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
There are three sets of dominoes of different colours. How can you put the dominoes from all three sets into a chain (according to the rules of the game) so that every two neighbouring dominoes are of a different colour?
How many ways can I schedule the first round of the Russian Football Championship, in which 16 teams are playing? (It is important to note who is the host team).
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.
Prove that there is no polyhedron that has exactly seven edges.
A straight corridor of length 100 m is covered with 20 rugs that have a total length of 1 km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?