KG-MO 2009

$a,b,c$ are sides of triangle $ABC$ . For any choosen triple from $(a + 1,b,c),(a,b + 1,c),(a,b,c + 1)$ there exist a triangle which sides are choosen triple. Find all possible values of area which triangle $ABC$ can take.

For function $f: \mathbb{R} \to \mathbb{R}$ given that $f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$ , calculate $f(2009)$ .

Does $a^2 + b^2 + c^2 \leqslant 2(ab + bc + ca)$ hold for every $a,b,c$ if it is known that $a^4 + b^4 + c^4 \leqslant 2(a^2 b^2 + b^2 c^2 + c^2 a^2 )$ .

Does there exist a function $f: {\Bbb N} \to {\Bbb N}$ such that $f(f(n - 1)) = f(n + 1) - f(n)$ for all $n > 2$ .

For any positive $a_1 ,a_2 ,...,a_n$ prove that $\frac {{a_1 }} {{a_2 + a_3 }} + \frac {{a_2 }} {{a_3 + a_4 }} + ... + \frac {{a_n }} {{a_1 + a_2 }} > \frac {n} {4}$ holds.