BALKAN 2006

Let $a$ , $b$ , $c$ be positive real numbers. Prove the inequality $$\frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}.$$

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$ , respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$ . The parallel lines from the points $A$ , $B$ , $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$ , $B_1$ and $C_1$ , respectively (apart from $A$ , $B$ , $C$ ). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point.

Greece

Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$ , $n+\frac 1{pm}$ , $p+\frac 1{mn}$ are integers.

Valentin Vornicu, Romania

Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$ , and $$a_{n+1} = \begin{cases} \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, a_n + m & \textrm{ otherwise. } \end{cases}$$ Find all values of $a$ such that the sequence is periodical (starting from the beginning).