This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 2

1991 Austrian-Polish Competition, 6
Tags: geometry , areas , area of a triangle , diagonal , bisect , convex
Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.5
Tags: geometry , bisect
A circle can be drawn around the quadrilateral $ABCD$. Let straight lines $AB$ and $CD$ intersect at point $M$, and straight lines $BC$ and $AD$ intersect at point $K$. (Points $B$ and $P$ lie on segments $AM$ and $AK$, respectively.) Let $P$ be the projection of point $M$ onto straight line $AK$, $L$ be the projection of point $K$ on the straight line $AM$. Prove that the straight line $LP$ divides the diagonal $BD$ in half.