KURSCHAK 1952

A circle $C$ touches three pairwise disjoint circles whose centers are collinear and none of which contains any of the others. Show that its radius must be larger than the radius of the middle of the three circles.

Show that if we choose any $n + 2$ distinct numbers from the set $\{1, 2, 3, . . . , 3n\}$ there will be two whose difference is greater than $n$ and smaller than $2n$ .

$ABC$ is a triangle. The point A' lies on the side opposite to $A$ and $BA'/BC = k$ , where $1/2 < k < 1$ . Similarly, $B'$ lies on the side opposite to $B$ with $CB'/CA = k$ , and $C'$ lies on the side opposite to $C$ with $AC'/AB = k$ . Show that the perimeter of $A'B'C'$ is less than $k$ times the perimeter of $ABC$ .