BALKAN-SL 2011

Given an integer $n \geq 3$ , determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition \begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}

Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$ . Prove, that \begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}

Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class $\mathcal{C}$ of the partition, the set of difference \begin{align*} \left{ \min { n \in \mathcal{C} \mid n >m } -m \mid m \in \mathcal{C} \right } \end{align*} be bounded?

Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$ , respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$ . The circles $ABC$ and $ALO$ meet again at point $N$ . Show that the angles $BAN$ and $CAM$ are equal.

Given a triangle $ABC$ , the line parallel to the side $BC$ and tangent to the incircle of the triangle meets the sides $AB$ and $AC$ at the points $A_1$ and $A_2$ , the points $B_1, B_2$ and $C_1, C_2$ are dened similarly. Show that $$AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)$$

Given an odd number $n >1$ , let \begin{align*} S ={ k \mid 1 \le k < n , \gcd(k,n) =1 } \end{align*} and let \begin{align*} T = { k \mid k \in S , \gcd(k+1,n) =1 } \end{align*} For each $k \in S$ , let $r_k$ be the remainder left by $\frac{k^{|S|}-1}{n}$ upon division by $n$ . Prove \begin{align*} \prod {k \in T} \left( r_k - r{n-k} \right) \equiv |S| ^{|T|} \pmod{n} \end{align*}

Let $n \in \mathbb{N}$ such that $p=17^{2n}+4$ is a prime. Show \begin{align*} p \mid 7^{\tfrac{p-1}{2}} +1 \end{align*}