65th Czech and Slovak Mathematical Olympiad

Let us consider the club $K$ with the least number of its members (in case there is more such clubs, we take any). We give to one of its members (let us call him Jacob) both a compass and a ruler. Each of the other members of the club will get a compass. Any other citizen will get a ruler. We show that this distribution complies with the conditions of the problem:

Any club which has Jacob as its member has certainly both instruments.

If there is a club where Jacob does not belong, then it has at least one common member with the club $K$, that is, there is at least a compass at disposal in the club. If there were no ruler in the club, it would mean that it is a “subclub” of $K$ and therefore has at least one member (Jacob) less than $K$, which is a contradiction with the choice of $K$. The described distribution really satisfies the conditions of the problem.