Olympiad problems

2004 Bosnia and Herzegovina Team Selection Test, 1

Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear

2018 Junior Regional Olympiad - FBH, 4

It is given $4$ circles in a plane and every one of them touches the other three as shown: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img] Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.

touching radius tangent circles geometry

1994 Bundeswettbewerb Mathematik, 3

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

cone touching geometry 3d geometry sphere number theory solid geometry