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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.

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