MK-MO 2008

Find all injective functions $f : \mathbb{N} \to \mathbb{N}$ which satisfy $$f(f(n)) \le\frac{n + f(n)}{2}$$ for each $n \in \mathbb{N}$ .

Positive numbers $a$ , $b$ , $c$ are such that $\left(a + b\right)\left(b + c\right)\left(c + a\right) = 8$ . Prove the inequality $$\frac {a + b + c}{3}\ge\sqrt [27]{\frac {a^3 + b^3 + c^3}{3}}$$

An acute triangle $ABC$ with $AB \neq AC$ is given. Let $V$ and $D$ be the feet of the altitude and angle bisector from $A$ , and let $E$ and $F$ be the intersection points of the circumcircle of $\triangle AVD$ with sides $AC$ and $AB$ , respectively. Prove that $AD$ , $BE$ and $CF$ have a common point.

We call an integer $n > 1$ good if, for any natural numbers $1 \le b_1, b_2, \ldots , b_{n-1} \le n - 1$ and any $i \in \{0, 1, \ldots , n - 1\}$ , there is a subset $I$ of $\{1, \ldots , n - 1\}$ such that $\sum_{k\in I} b_k \equiv i \pmod n$ . (The sum over the empty set is zero.) Find all good numbers.