IMO 1966 Problem 6

Let $ABC$ be a triangle, and let $P$ , $Q$ , $R$ be three points in the interiors of the sides $BC$ , $CA$ , $AB$ of this triangle. Prove that the area of at least one of the three triangles $AQR$ , $BRP$ , $CPQ$ is less than or equal to one quarter of the area of triangle $ABC$ .

Alternative formulation: Let $ABC$ be a triangle, and let $P$ , $Q$ , $R$ be three points on the segments $BC$ , $CA$ , $AB$ , respectively. Prove that

$\min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$ ,

where the abbreviation $\left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $P_1P_2P_3$ .