KURSCHAK 1954

$ABCD$ is a convex quadrilateral with $AB + BD = AC + CD$ . Prove that $AB < AC$ .

Every planar section of a three-dimensional body $B$ is a disk. Show that B must be a ball.

A tournament is arranged amongst a finite number of people. Every person plays every other person just once and each game results in a win to one of the players (there are no draws). Show that there must a person $X$ such that, given any other person $Y$ in the tournament, either $X$ beat $Y$ , or $X$ beat $Z$ and $Z$ beat $Y$ for some $Z$ .