BALKAN-SL 2009

Let $N \in \mathbb{N}$ and $x_k \in [-1,1]$ , $1 \le k \le N$ such that $\sum_{k=1}^N x_k =s$ . Find all possible values of $\sum_{k=1}^N |x_k|$

Let $ABCD$ be a square and points $M$ $\in$ $BC$ , $N \in CD$ , $P$ $\in$ $DA$ , such that $\angle BAM$ $=$ $x$ , $\angle CMN$ $=$ $2x$ , $\angle DNP$ $=$ $3x$ [list=i] [] Show that, for any $x \in (0, \tfrac{\pi}{8} )$ , such a configuration exists [] Determine the number of angles $x \in ( 0, \tfrac{\pi}{8} )$ for which $\angle APB =4x$

Denote by $S(x)$ the sum of digits of positive integer $x$ written in decimal notation. For $k$ a fixed positive integer, define a sequence $(x_n)_{n \geq 1}$ by $x_1=1$ and $x_{n+1}$ $=$ $S(kx_n)$ for all positive integers $n$ . Prove that $x_n$ $<$ $27 \sqrt{k}$ for all positive integer $n$ .

Given the monic polynomial \begin{align*} P(x) = x^N +a_{N-1}x^{N-1} + \ldots + a_1 x + a_0 \in \mathbb{R}[x] \end{align*} of even degree $N$ $=$ $2n$ and having all real positive roots $x_i$ , for $1 \le i \le N$ . Prove, for any $c$ $\in$ $[0, \underset{1 \le i \le N}{\min} \{x_i \} )$ , the following inequality \begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}

We denote the set of nonzero integers and the set of non-negative integers with $\mathbb Z^*$ and $\mathbb N_0$ , respectively. Find all functions $f:\mathbb Z^* \to \mathbb N_0$ such that: $a)$ $f(a+b)\geq min(f(a), f(b))$ for all $a,b$ in $\mathbb Z^*$ for which $a+b$ is in $\mathbb Z^*$ . $b)$ $f(ab)=f(a)+f(b)$ for all $a,b$ in $\mathbb Z^*$ .

Let $n\geq 2$ be a positive integer and \begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*} be a polynomial with integer coefficients, such that $\mid c_n \mid$ is a prime number and \begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*} Prove that the polynomial $P(X)$ is irreducible in the $\mathbb{Z}[x]$

For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} [list=a] [] Prove that $K_m \geq 0$ for every odd positive integer $m$ [] Let $M$ $= \prod_{cyc} (x-y)^2$ . Prove, $K_7+M^2 K_1 \geq M K_4$

Let $A_1, A_2, \ldots , A_m$ be subsets of the set $\{ 1,2, \ldots , n \}$ , such that the cardinal of each subset $A_i$ , such $1 \le i \le m$ is not divisible by $30$ , while the cardinal of each of the subsets $A_i \cap A_j$ for $1 \le i,j \le m$ , $i \neq j$ is divisible by $30$ . Prove \begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor \le 3n \end{align*}

In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$ , and $\angle ADB = 45^o$ . Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$ . Point $Q$ lies on the ray $DP$ such that $DQ = DK$ . The perpendicular at $P$ to $AC$ meets $KD$ at $L$ . Prove that $PL^2 = DQ \cdot PQ$ .

Let $ABCD$ be a convex quadrilateral, and $P$ be a point in its interior. The projections of $P$ on the sides of the quadrilateral lie on a circle with center $O$ . Show that $O$ lies on the line through the midpoints of $AC$ and $BD$ .

Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$ . Prove that $(AECZ) = (EBZD) = (ABCD)$ .

Determine all integers $1 \le m, 1 \le n \le 2009$ , for which \begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}