BALKAN-SL 2013

Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$ . Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$

Let $a, b, c$ and $d$ are positive real numbers so that $abcd = \frac14$ . Prove that holds $$\left( 16ac +\frac{a}{c^2b}+\frac{16c}{a^2d}+\frac{4}{ac}\right)\left( bd +\frac{b}{256d^2c}+\frac{d}{b^2a}+\frac{1}{64bd}\right) \ge \frac{81}{4}$$ When does the equality hold?

Prove that the polynomial $P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1$ , $n \in N^*$ , cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to $1$ .

Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$

Determine all positive integers $n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$ , as polynomials in $x, y, z$ with integer coefficients.

Suppose that $k$ is a positive integer. A bijective map $f : Z \to Z$ is said to be $k$ -jumpy if $|f(z) - z| \le k$ for all integers $z$ . Is it that case that for every $k$ , each $k$ -jumpy map is a composition of $1$ -jumpy maps? It is well known that this is the case when the support of the map is finite.

Some squares of an $n \times n$ chessboard have been marked ( $n \in N^*$ ). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$ , then there exists a rectangle whose vertices are centers of marked squares.

The square $ABCD$ is divided into $n^2$ equal small (elementary) squares by parallel lines to its sides, (see the figure for the case $n = 4$ ). A spider starts from point $A$ and moving only to the right and up tries to arrive at point $C$ . Every " movement" of the spider consists of: " $k$ steps to the right and $m$ steps up" or " $m$ steps to the right and $k$ steps up" (which can be performed in any way). The spider first makes $l$ "movements" and in then, moves to the right or up without any restriction. If $n = m \cdot l$ , find all possible ways the spider can approach the point $C$ , where $n, m, k, l$ are positive integers with $k < m$ .

A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board's squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.

The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A proper path of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$ . Show that there exists a proper path of length $n$ .

Let $ABCD$ be a quadrilateral, let $O$ be the intersection point of diagonals $AC$ and $BD$ , and let $P$ be the intersection point of sides $AB$ and $CD$ . Consider the parallelograms $AODE$ and $BOCF$ . Prove that $E, F$ and $P$ are collinear.

Two circles $\Gamma_1$ and $\Gamma_2$ intersect at points $M,N$ . A line $\ell$ is tangent to $\Gamma_1 ,\Gamma_2$ at $A$ and $B$ , respectively. The lines passing through $A$ and $B$ and perpendicular to $\ell$ intersects $MN$ at $C$ and $D$ respectively. Prove that $ABCD$ is a parallelogram.

Let $c(O, R)$ be a circle, $AB$ a diameter and $C$ an arbitrary point on the circle different than $A$ and $B$ such that $\angle AOC > 90^o$ . On the radius $OC$ we consider point $K$ and the circle $c_1(K, KC)$ . The extension of the segment $KB$ meets the circle $(c)$ at point $E$ . From $E$ we consider the tangents $ES$ and $ET$ to the circle $(c_1)$ . Prove that the lines $BE, ST$ and $AC$ are concurrent.

Let $ABC$ be an acute triangle with $AB < AC < BC$ inscribed in a circle $(c)$ and let $E$ be an arbitrary point on its altitude $CD$ . The circle $(c_1)$ with diameter $EC$ , intersects the circle $(c)$ at point $K$ (different than $C$ ), the line $AC$ at point $L$ and the line $BC$ at point $M$ . Finally the line $KE$ intersects $AB$ at point $N$ . Prove that the quadrilateral $DLMN$ is cyclic.

Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$

Determine all quadruplets ( $x, y, z, t$ ) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$ .

Let $p$ be a prime number greater than $3$ . Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$ .

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

Two distinct positive integers are called close if their greatest common divisor equals their difference. Show that for any $n$ , there exists a set $S$ of $n$ elements such that any two elements of $S$ are close.

Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$ , if and only if $a-b$ divides $2ab$ .

Let $n\ge 2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that $x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$