MK-MO 2007

Let $a, b, c$ be positive real numbers. Prove that $$1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.$$

In a trapezoid $ABCD$ with a base $AD$ , point $L$ is the orthogonal projection of $C$ on $AB$ , and $K$ is the point on $BC$ such that $AK$ is perpendicular to $AD$ . Let $O$ be the circumcenter of triangle $ACD$ . Suppose that the lines $AK , CL$ and $DO$ have a common point. Prove that $ABCD$ is a parallelogram.

Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy $$a | b+c+bc, b | c+a+ca, c | a+b+ab.$$ Prove that at least one of the numbers $a, b, c$ is not prime.

Find all functions $f : \mathbb{R}\to\mathbb{R}$ that satisfy $$f (x^{3} + y^{3}) = x^{2}f (x) + yf (y^{2})$$ for all $x, y \in\mathbb R.$

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