KURSCHAK 1961

Given any four distinct points in the plane, show that the ratio of the largest to the smallest distance between two of them is at least $\sqrt2$ .

$x, y, z$ are positive reals less than $1$ . Show that at least one of $(1 - x)y$ , $(1 - y)z$ and $(1 - z)x$ does not exceed $\frac14$ .

Two circles centers $O$ and $O'$ are disjoint. $PP'$ is an outer tangent (with $P$ on the circle center O, and P' on the circle center $O'$ ). Similarly, $QQ'$ is an inner tangent (with $Q$ on the circle center $O$ , and $Q'$ on the circle center $O'$ ). Show that the lines $PQ$ and $P'Q'$ meet on the line $OO'$ .