Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at a point P. Let $M,N$ be the midpoints of the sides $AB$ and $CD$ respectively. Prove that the area of the triangle $PMN$ is equal to the quarter of the absolute value of the difference between the area of the triangle $DAP$ and the area of the triangle $BCP$: \[ \text{area}(\triangle MNP)=\frac{1}{4}\left|\text{area}(\triangle DAP)-\text{area}(\triangle BCP)\right |.\] We received only one solution, from Sasha Aksenchuk. Sasha's solution uses analytic geometry and is similar to one of our in-house solutions. For a complete solution see the following link {{:pow:2024sproblem6.pdf|Solution}}.