KURSCHAK 1980

The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.

Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$ .

In a certain country there are two tennis clubs consisting of $1000$ and $1001$ members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within $11$ games, who is in the $1001$ st place among the $2001$ players in these clubs. It is assumed that a stronger player always beats a weaker one.