RO-NMO 2015 Problem 93

Let be a point $P$ in the interior of a triangle $ABC.$ The lines $AP,BP,CP$ meet $BC,AC,$ respectively, $AB$ at $A_1,B_1,$ respectively, $C_1.$ If $$\mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} ,$$ show that $P$ lies on a median of $ABC.$ $\mathcal{A}$ denotes area.