Math contest proof problem Could someone help me with this?
Let $x,y,z,w$ be positive real numbers such that $x + y + z = w.$ Show that $${(w−x)(w−y)(w−z) \over (w + x)(w + y)(w + z)} \le \frac 18$$

Use the substitution $a = y+z, b = z+x, c = x+y$, then the inequality is
$$ \frac{ abc} {(a+b)(b+c)(c+a) } \leq \frac{1}{8}. $$
Cross multiply and apply $AM-GM$ directly to the terms, to get
$$ (a+b)(b+c)(c+a) \geq 2 \sqrt{ab} \times 2 \sqrt{bc} \times 2 \sqrt{ca} = 8abc$$
