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.
All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
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?
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).
A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?
A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).