Olympiad problems

1991 Bundeswettbewerb Mathematik, 3

A set $M$ of points in the plane will be called obtuse, if any 3 points from $M$ are the vertices of an obtuse triangle. a.) Prove: For each finite obtuse set $M$ there is a point in the plane with the following property: $P$ is no element from $M$ and $M \cup \{P\}$ is also obtuse. b.) Determine whether the statement from a.) will remain valid, if it is replaced by infinite.

2003 Federal Math Competition of S&M, Problem 3

Tags: geometry , rectangle , symmetry , circumcircle , perpendicular bisector , geometry solved

Let $ABCD$ be a rectangle. Determine the set of all points $P$ from the region between the parallel lines $AB$ and $CD$ such that $\angle APB=\angle CPD$.

2020 Romania EGMO TST, P1

Tags: geometry , Poland , geometry solved , radical axis , Angle Chasing

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

2004 China Team Selection Test, 3

Tags: geometry , perimeter , trigonometry , geometry solved

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.