CMO 1995

Let $f(x)=\frac{9^x}{9^x + 3}$ . Evaluate $\sum_{i=1}^{1995}{f\left(\frac{i}{1996}\right)}$ .

Let $\{a,b,c\}\in \mathbb{R}^{+}$ . Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$ .

Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than $180^{\circ}$ . Let $C$ be a convex polygon with $s$ sides. The interior region of $C$ is the union of $q$ quadrilaterals, none of whose interiors overlap each other. $b$ of these quadrilaterals are boomerangs. Show that $q\ge b+\frac{s-2}{2}$ .

Let $n$ be a constant positive integer. Show that for only non-negative integers $k$ , the Diophantine equation $\sum_{i=1 }^{n}{ x_i ^3}=y^{3k+2}$ has infinitely many solutions in the positive integers $x_i, y$ .

$u$ is a real parameter such that $0<u<1$ . For $0\le x \le u$ , $f(x)=0$ . For $u\le x \le n$ , $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$ . The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$ . Show that there exists a positive integer $k$ for which $u_k=0$ .