CGMO 2003 Problem 1

Let $ABC$ be a triangle. Points $D$ and $E$ are on sides $AB$ and $AC,$ respectively, and point $F$ is on line segment $DE.$ Let $\frac {AD}{AB} = x,$ $\frac {AE}{AC} = y,$ $\frac {DF}{DE} = z.$ Prove that

(1) $S_{\triangle BDF} = (1 - x)y S_{\triangle ABC}$ and $S_{\triangle CEF} = x(1 - y) (1 - z)S_{\triangle ABC};$

(2) $\sqrt [3]{S_{\triangle BDF}} + \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$