Olympiad problems

1967 Dutch Mathematical Olympiad, 3

The convex pentagon $ABC DE$ is given, such that $AB,BC,CD$ and $DE$ are parallel to one of the diagonals. Prove that this also applies to $EA$.

pentagon geometry parallel

1993 Mexico National Olympiad, 3

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

pentagon geometry area of a triangle combinatorial geometry

2012 Korea Junior Math Olympiad, 2

A pentagon $ABCDE$ is inscribed in a circle $O$, and satisfies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.

pentagon concyclic geometry angle chasing

1962 All Russian Mathematical Olympiad, 020

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

pentagon minimum geometry maximum