Problem

Source: 45th International Tournament of Towns, Senior A-Level P4, Fall 2023

Tags: geometry



A convex quadrilateral $A B C D$ with area of $S$ is given. Inside each side of the quadrilateral a point is selected. These points are consecutively linked by segments, so that $A B C D$ is split into a smaller quadrilateral and 4 triangles. Prove that the area of at least one triangle does not exceed $S / 8$. Mikhail Malkin