BALKAN-SL 2008

For all $\alpha_1, \alpha_2,\alpha_3 \in \mathbb{R}^+$ , Prove \begin{align*} \sum \frac{1}{2\nu \alpha_1 +\alpha_2+\alpha_3} > \frac{2\nu}{2\nu +1} \left( \sum \frac{1}{\nu \alpha_1 + \nu \alpha_2 + \alpha_3} \right) \end{align*} for every positive real number $\nu$

Let $(a_m)$ be a sequence satisfying $a_n \geq 0$ , $n=0,1,2,\ldots$ Suppose there exists $A >0$ , $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$ . Prove that there exists $B>0$ such that \begin{align*} a_n \le \frac{B}{n} \text{for }1 \le n \end{align*}

We consider the set \begin{align*} \mathbb{C}^{\nu} = { (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} }, \nu \geq 2 \end{align*} and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the $\nu-$ tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$ th roots of $-1$ , where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \mu =0,1, \ldots , \nu -1 \end{align*} Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ , we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine [list=i] [] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right)$ have measures less than or equal to $1$ [] for $\nu =4$ , the minimal value of $\kappa \in \mathbb{N}$ , for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} 1 \le i \le 4 \end{align*}

Consider an integer $n \geq 1$ , $a_1,a_2, \ldots , a_n$ real numbers in $[-1,1]$ satisfying \begin{align*}a_1+a_2+\ldots +a_n=0 \end{align*} and a function $f: [-1,1] \mapsto \mathbb{R}$ such \begin{align*} \mid f(x)-f(y) \mid \le \mid x-y \mid \end{align*} for every $x,y \in [-1,1]$ . Prove \begin{align*} \left| f(x) - \frac{f(a_1) +f(a_2) + \ldots + f(a_n)}{n} \right| \le 1 \end{align*} for every $x$ $\in [-1,1]$ . For a given sequence $a_1,a_2, \ldots ,a_n$ , Find $f$ and $x$ so hat the equality holds.

Prove that if $x,y,z \in \mathbb{R}^+$ such that $xy,yz,zx$ are sidelengths of a triangle and $k$ $\in$ $[-1,1]$ , then \begin{align*} \sum \frac{\sqrt{xy}}{\sqrt{xz+yz+kxy}} \geq 2 \sqrt{1-k} \end{align*} Determine the equality condition too.

Let $x,y,z,t \in \mathbb{R}_{\geq 0}$ . Show \begin{align*} \sqrt{xy}+\sqrt{xz}+\sqrt{xt}+\sqrt{yz}+\sqrt{yt}+\sqrt{zt} \geq 3 \sqrt[3]{xyz+xyt+xzt+yzt} \end{align*} and determine the equality cases.

All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$ . One day, Professor $D$ came up with a polynomial with real coefficients and power $n$ . Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$ th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$ th office he wrote $2^{n+2}$ $-n-3$ . [list=a] [] Prove that Professor D made a calculation error [] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them. [/list]

In one of the countries, there are $n \geq 5$ cities operated by two airline companies. Every two cities are operated in both directions by at most one of the companies. The government introduced a restriction that all round trips that a company can offer should have atleast six cities. Prove that there are no more than $\lfloor \tfrac{n^2}{3} \rfloor$ flights offered by these companies.

An array $n \times n$ is given, consisting of $n^2$ unit squares. A pawn is placed arbitrarily on a unit square. A move of the pawn means a jump from a square of the $k$ th column to any square of the $k$ th row. Show that there exists a sequence of $n^2$ moves of the pawn so that all the unit squares of the array are visited once and, in the end, the pawn returns to the original position.

In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$ . The perpendicular bisectors of $GA, GB$ meet at $C_1$ ,of $GB,GC$ meet at $A _1$ , and $GC,GA$ meet at $B_1$ . Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$ .

The circle $k_a$ touches the extensions of sides $AB$ and $BC$ , as well as the circumscribed circle of the triangle $ABC$ (from the outside). We denote the intersection of $k_a$ with the circumscribed circle of the triangle $ABC$ by $A'$ . Analogously, we define points $B'$ and $C'$ . Prove that the lines $AA',BB'$ and $CC'$ intersect in one point.

On triangle $ABC$ the $AM$ ( $M\in BC$ ) is median and $BB_1$ and $CC_1$ ( $B_1 \in AC,C_1 \in AB$ ) are altitudes. The stright line $d$ is perpendicular to $AM$ at the point $A$ and intersect the lines $BB_1$ and $CC_1$ at the points $E$ and $F$ respectively. Let denoted with $\omega$ the circle passing through the points $E, M$ and $F$ and with $\omega_1$ and with $\omega_2$ the circles that are tangent to segment $EF$ and with $\omega$ at the arc $EF$ which is not contain the point $M$ . If the points $P$ and $Q$ are intersections points for $\omega_1$ and $\omega_2$ then prove that the points $P, Q$ and $M$ are collinear.

In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$ , $M$ and $N$ are the midpoints of the segments $[BC]$ , respectively $[XY]$ , and the straight lines $XY$ and $BC$ meet in $K$ . Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$ .

Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

Prove that for every natural number $a$ , there exists a natural number that has the number $a$ (the sequence of digits that constitute $a$ ) at its beginning, and which decreases $a$ times when $a$ is moved from its beginning to it end (any number zeros that appear in the beginning of the number obtained in this way are to be removed).

Example [list=i] [] $a=4$ , then $\underline{4}10256= 4 \cdot 10256\underline{4}$ [] $a=46$ , then $\underline{46}0100021743857360295716= 46 \cdot 100021743857360295716\underline{46}$

The sequence $(\chi_n) _{n=1}^{\infty}$ is defined as follows \begin{align*} \chi_{n+1}=\chi_n + \chi _{\lceil \frac{n}{2} \rceil} ~, \chi_1 =1 \end{align*} Prove that none of the terms of this sequence are divisible by $4$

Let $(a_n)$ be a sequence with $a_1=0$ and $a_{n+1}=2+a_n$ for odd $n$ and $a_{n+1}=2a_n$ for even $n$ . Prove that for each prime $p>3$ , the number \begin{align*} b=\frac{2^{2p}-1}{3} \mid a_n \end{align*} for infinitely many values of $n$

Let $(x_n)$ , $n=1,2, \ldots$ be a sequence defined by $x_1=2008$ and \begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n ~ ~ ~ \forall n \geq 2 \end{align*} Let the sequence $a_n=x_n + \frac{1}{n} S_n$ , $n=1,2,3, \ldots$ where $S_n$ $=$ $x_1+x_2 +\ldots +x_n$ . Determine the values of $n$ for which the terms of the sequence $a_n$ are perfect squares of an integer.