# MMO 2007 

1 Let $a, b, c$ be positive real numbers. Prove that

$$
1+\frac{3}{a b+b c+c a} \geq \frac{6}{a+b+c}
$$

2 In a trapezoid $A B C D$ with a base $A D$, point $L$ is the orthogonal projection of $C$ on $A B$, and $K$ is the point on $B C$ such that $A K$ is perpendicular to $A D$. Let $O$ be the circumcenter of triangle $A C D$. Suppose that the lines $A K, C L$ and $D O$ have a common point. Prove that $A B C D$ is a parallelogram.

3 Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy

$$
a|b+c+b c, b| c+a+c a, c \mid a+b+a b
$$

Prove that at least one of the numbers $a, b, c$ is not prime.

4 Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy

$$
f\left(x^{3}+y^{3}\right)=x^{2} f(x)+y f\left(y^{2}\right)
$$

for all $x, y \in \mathbb{R}$.

5 Let $n$ be a natural number divisible by 4 . Determine the number of bijections $f$ on the set $\{1,2, \ldots, n\}$ such that $f(j)+f^{-1}(j)=n+1$ for $j=1, \ldots, n$.