TR-MO 2003 Problem 2

Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$ , respectively. Show that, $$\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}$$ where $s_1=\text{Area}(AKN)$ , $s_2=\text{Area}(BKL)$ , $s_3=\text{Area}(CLM)$ , $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$ .