KMO 1995

For any positive integer $m$ ,show that there exist integers $a,b$ satisfying $\left | a \right |\leq m$ , $\left | b \right |\leq m$ , $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$

find all functions from the nonegative integers into themselves, such that: $2f(m^2+n^2)=f^2(m)+f^2(n)$ and for $m\geq n$ $f(m^2)\geq f(n^2)$ .

Let $ABC$ be an equilateral triangle of side $1$ , $D$ be a point on $BC$ , and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$ . Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$ .