BALKAN 2012

$Let$A, B$and$C$be points lying on a circle$\Gamma$with centre$O$. Assume that$\angle A B C>90$. Let$D$be the point of intersection of the line$A B$with the line perpendicular to$A C$at$C$. Let$l$be the line through$D$which is perpendicular to$A O$. Let$E$be the point of intersection of$l$with the line$A C$, and let$F$be the point of intersection of$\Gamma$with$l$that lies between$D$and$E$. Prove that the circumcircles of triangles$B F E$and$C F D$are tangent at$F$.

Prove that

$$\sum_{c y c}(x+y) \sqrt{(z+x)(z+y)} \geq 4(x y+y z+z x),$$

for all positive real numbers $x, y$ and $z$.

Let $n$ be a positive integer. Let $P_{n}=\left\{2^{n}, 2^{n-1} \cdot 3,2^{n-2} \cdot 3^{2}, \ldots, 3^{n}\right\}$. For each subset $X$ of $P_{n}$, we write $S_{X}$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}$. Prove that there is a subset $Y$ of $P_{n}$ such that $0 \leq y-S_{Y}<2^{n}$

$Let$\mathbb{Z}^{+}$be the set of positive integers. Find all functions$f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$such that the following conditions both hold: (i)$f(n!)=f(n)$! for every positive integer$n$, (ii)$m-n$divides$f(m)-f(n)$whenever$m$and$n$ are different positive integers.