KMO 2007

For the vertex $A$ of a triangle $ABC$ , let $l_a$ be the distance between the projections on $AB$ and $AC$ of the intersection of the angle bisector of ∠ $A$ with side $BC$ . Define $l_b$ and $l_c$ analogously. If $l$ is the perimeter of triangle $ABC$ , prove that $\frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$ .

Let f:N→N be a function satisfying $kf(n)\le f(kn)\le kf(n) + k - 1$ for all $k, n\in N$ .

(a)Prove that $f(a) + f(b)\le f(a + b)\le f(a) + f(b) + 1$ for all $a, b\in N$ .

(b)If $f$ satisfies $f(2007n)\le 2007f(n) + 200$ for every $n\in N$ , show that there exists $c\in N$ such that $f(2007c) = 2007f(c)$ .