Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Balkan MO Shortlist
Let $I$ be the incenter of $\triangle ABC$ and let $H_a$, $H_b$, and $H_c$ be the orthocenters of $\triangle BIC$ , $\triangle CIA$, and $\triangle AIB$, respectively. The lines $H_aH_b$ meets $AB$ at $X$ and the line $H_aH_c$ meets $AC$ at $Y$. If the midpoint $T$ of the median $AM$ of $\triangle ABC$ lies on $XY$, prove that the line $H_aT$ is perpendicular to $BC$
2014 Balkan MO Shortlist
$\boxed{N1}$Let $n$ be a positive integer,$g(n)$ be the number of positive divisors of $n$ of the form $6k+1$ and $h(n)$ be the number of positive divisors of $n$ of the form $6k-1,$where $k$ is a nonnegative integer.Find all positive integers $n$ such that $g(n)$ and $h(n)$ have different parity.
2014 Balkan MO Shortlist
$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if

$$x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}$$for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$
2014 Balkan MO Shortlist
$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$
2014 Balkan MO Shortlist
$\boxed{N5}$Let $a,b,c,p,q,r$ be positive integers such that $a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.$

Prove that $a=b=c$ or $p=q=r.$
2014 Balkan MO Shortlist
Let $ f: \mathbb{N} \rightarrow \mathbb{N} $ be a function from the positive integers to the positive integers for which $ f(1)=1,f(2n)=f(n) $ and $ f(2n+1)=f(n)+f(n+1) $ for all $ n\in \mathbb{N} $. Prove that for any natural number $ n $, the number of odd natural numbers $ m $ such that $ f(m)=n $ is equal to the number of positive integers not greater than $ n $ having no common prime factors with $ n $.
2013 Balkan MO Shortlist
Positive real numbers $a, b,c$ satisfy $ab + bc+ ca = 3$. Prove the inequality $$\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}$$
2013 Balkan MO Shortlist
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?
2013 Balkan MO Shortlist
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 $1$.
2013 Balkan MO Shortlist
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 $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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 $k < m$.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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 $n$.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
Let $p$ be a prime number. Determine all triples $(a,b,c)$ of positive integers such that $a + b + c < 2p\sqrt{p}$ and

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}$
2013 Balkan MO Shortlist
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.
2013 Balkan MO Shortlist
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 $p^2$.
2013 Balkan MO Shortlist
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$
2013 Balkan MO Shortlist
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$
2013 Balkan MO Shortlist
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.
2013 Balkan MO Shortlist
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 $2ab$.
2013 Balkan MO Shortlist
Let $n\ge  2$ be a given integer. Determine all sequences $x_1,...,x_n$ of positive rational numbers such that

$x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}$
2012 Balkan MO Shortlist
Let $a,b,c\ge 0$ and $a+b+c=\sqrt2$. Show that

$$\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}$$



Click to reveal hidden textIn general if $a_1, a_2, \cdots , a_n \ge 0$ and $\sum_{i=1}^n a_i=\sqrt2$ we have

$$\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}$$
2012 Balkan MO Shortlist
Determine the maximum possible number of distinct real roots of a polynomial $P(x)$ of degree $2012$ with real coefficients satisfying the condition

\begin{align*}  P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*}for all real numbers $a,b,c \in \mathbb{R}$ with $a+b+c=0$
2012 Balkan MO Shortlist
Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$
2012 Balkan MO Shortlist
Let $ABC$ be a triangle, and let $\ell$ be the line passing through the circumcenter of $ABC$ and parallel to the bisector of the angle $\angle A$. Prove that the line $\ell$  passes through the orthocenter of $ABC$ if and only if $AB = AC$ or $\angle BAC = 120^o$
2012 Balkan MO Shortlist
Let $ABC$ be a triangle with circumcircle $c$ and circumcenter $O$, and let $D$ be a point on the side $BC$ different from the vertices and the midpoint of $BC$. Let $K$ be the point where the circumcircle $c_1$ of the triangle $BOD$ intersects $c$ for the second time and let $Z$ be the point where $c_1$ meets the line $AB$. Let $M$ be the point where the circumcircle $c_2$ of the triangle $COD$ intersects $c$ for the second time and let $E$ be the point where $c_2$ meets the line $AC$. Finally let $N$ be the point where the circumcircle $c_3$ of the triangle $AEZ$ meets $c$ again. Prove that the triangles $ABC$ and $NKM$ are congruent.
2012 Balkan MO Shortlist
Let $M$ be the point of intersection of the diagonals of a cyclic quadrilateral $ABCD$. Let $I_1$ and $I_2$ are the incenters of triangles $AMD$ and $BMC$, respectively, and let $L$ be the point of intersection of the lines $DI_1$ and $CI_2$. The foot of the perpendicular from the midpoint $T$ of $I_1I_2$ to $CL$ is $N$, and $F$ is the midpoint of $TN$. Let $G$ and $J$ be the points of intersection of the line $LF$ with $I_1N$ and $I_1I_2$, respectively. Let $O_1$ be the circumcenter of triangle $LI_1J$, and let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $O_1L$ and $O_1J$, respectively. Let $V$ and $S$ be the second points of intersection of $I_1O_1$ with $\Gamma_1$ and $\Gamma_2$, respectively. If $K$ is point where the circles $\Gamma_1$ and $\Gamma_2$ meet again, prove that $K$ is the circumcenter of the triangle $SVG$.
2012 Balkan MO Shortlist
$\boxed{\text{G5}}$ The incircle of a triangle $ABC$ touches its sides $BC$,$CA$,$AB$ at the points $A_1$,$B_1$,$C_1$.Let the projections of the orthocenter $H_1$ of the triangle $A_{1}B_{1}C_{1}$ to the lines $AA_1$ and $BC$ be $P$ and $Q$,respectively. Show that $PQ$ bisects the line segment $B_{1}C_{1}$
2012 Balkan MO Shortlist
Let $P$ and $Q$ be points inside a triangle $ABC$ such that $\angle PAC = \angle QAB$ and $\angle PBC = \angle QBA$. Let $D$ and $E$ be the feet of the perpendiculars from $P$ to the lines $BC$ and $AC$, and $F$ be the foot of perpendicular from $Q$ to the line $AB$. Let $M$ be intersection of the lines $DE$ and $AB$. Prove that $MP \perp CF$
2012 Balkan MO Shortlist
$ABCD$ is a cyclic quadrilateral. The lines $AD$ and $BC$ meet at X, and the lines $AB$ and $CD$ meet at $Y$ . The line joining the midpoints $M$ and $N$ of the diagonals $AC$ and $BD$, respectively, meets the internal bisector of angle $AXB$ at $P$ and the external bisector of angle $BYC$ at $Q$. Prove that $PXQY$ is a rectangle
2012 Balkan MO Shortlist
A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.
2012 Balkan MO Shortlist
Let the sequences $(a_n)_{n=1}^{\infty}$  and $(b_n)_{n=1}^{\infty}$  satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$.

Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.
2011 Balkan MO Shortlist
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*}
2011 Balkan MO Shortlist
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*}
2011 Balkan MO Shortlist
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.
2011 Balkan MO Shortlist
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 dened similarly. Show that

$$AA_1 \cdot  AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2  \ge \frac19 (AB^2 + BC^2 + CA^2)$$
2011 Balkan MO Shortlist
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?
2011 Balkan MO Shortlist
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*}
2011 Balkan MO Shortlist
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*}
2010 Balkan MO Shortlist
Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$

\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}
2010 Balkan MO Shortlist
Let $a,b,c,d$ be positive real numbers. Prove that

$$(\frac{a}{a+b})^{5}+(\frac{b}{b+c})^{5}+(\frac{c}{c+d})^{5}+(\frac{d}{d+a})^{5}\ge \frac{1}{8}$$
2010 Balkan MO Shortlist
Let $n>2$ be a positive integer. Consider all numbers $S$ of the form

\begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*}with $k>1$ and $a_i$ begin positive integers such that $a_1+a_2+ \ldots + a_k=n$. Determine all the numbers that can be represented in the given form.
2010 Balkan MO Shortlist
In a soccer tournament each team plays exactly one game with all others. The winner gets $3$ points, the loser gets $0$ and each team gets $1$ point in case of a draw. It is known that $n$ teams ($n \geq 3$) participated in the tournament and the final classification is given by the arithmetical progression of the points, the last team having only 1 point.



Prove that this configuration is unattainable when $n=12$

Find all values of $n$ and all configurations when this is possible
2010 Balkan MO Shortlist
A grasshopper jumps on the plane from an integer point (point with integer coordinates) to another integer point according to the following rules: His first jump is of length $\sqrt{98}$, his second jump is of length $\sqrt{149}$, his next jump is of length $\sqrt{98}$, and so on, alternatively. What is the least possible odd number of moves in which the grasshopper could return to his starting point?
2010 Balkan MO Shortlist
Integers are written in the cells of a table $2010 \times 2010$. Adding $1$ to all the numbers in a row or in a column is called a move. We say that a table is equilibrium if one can obtain after finitely many moves a table in which all the numbers are equal.



Find the largest positive integer $n$, for which there exists an equilibrium table containing the numbers $2^0, 2^1, \ldots , 2^n$.

For this $n$, find the maximal number that may be contained in such a table.
2010 Balkan MO Shortlist
A train consist of $2010$ wagons containing gold coins, all of the same shape. Any two coins have equal weight provided that they are in the same wagon, and differ in weight if they are in different ones. The weight of a coin is one of the positive reals

\begin{align*} m_1 <m_2 <\ldots <m_{2010} \end{align*}Each wagon is marked by a label with one of the numbers $m_1,m_2, \ldots , m_{2010}$ (the numbers on different labels are different).



A controller has a pair of scales (allowing only to compare masses) at his disposal. During each measurement he can use an arbitrary number of coins from any of the wagons. The controller has the task to establish: if all labels show rightly the common weight of the coins in a wagon or if there exists at least one wrong label. What is the least number of measurement that the controller has to perform to accomplish his task?
2010 Balkan MO Shortlist
Let $ABCDE$ be a pentagon with $\hat{A}=\hat{B}=\hat{C}=\hat{D}=120^{\circ}$. Prove that $4\cdot AC \cdot BD\geq 3\cdot AE \cdot ED$.
2010 Balkan MO Shortlist
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle
2010 Balkan MO Shortlist
The incircle of a triangle $A_0B_0C_0$ touches the sides $B_0C_0,C_0A_0,A_0B_0$ at the points $A,B,C$ respectively, and the incircle of the triangle $ABC$ with incenter $ I$ touches the sides $BC,CA, AB$ at the points $A_1, B_1,C_1$, respectively. Let $\sigma(ABC)$ and $\sigma(A_1B_1C)$ be the areas of the triangles $ABC$ and $A_1B_1C$ respectively. Show that if $\sigma(ABC) = 2 \sigma(A_1B_1C)$ , then the lines $AA_0, BB_0,IC_1$ pass through a common point .
2010 Balkan MO Shortlist
Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$.

(a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$.

(b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.
2010 Balkan MO Shortlist
In a triangle $ABC$ the excircle at the side $BC$ touches $BC$ in point $D$ and the lines $AB$ and $AC$ in points $E$ and $F$ respectively. Let $P$ be the projection of $D$ on $EF$. Prove that the circumcircle $k$ of the triangle $ABC$ passes through $P$ if and only if $k$ passes through the midpoint $M$ of the segment $EF$.
2010 Balkan MO Shortlist
A triangle $ABC$ is given. Let $M$ be the midpoint of the side $AC$ of the triangle and $Z$ the image of point $B$ along the line $BM$. The circle with center $M$ and radius $MB$ intersects the lines $BA$ and $BC$ at the points $E$ and $G$ respectively. Let $H$ be the point of intersection of $EG$ with the line $AC$, and $K$ the point of intersection of $HZ$ with the line $EB$. The perpendicular from point $K$ to the line $BH$  intersects the lines $BZ$ and $BH$ at the points $L$ and $N$, respectively.

If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that, the lines $BZ, KN$ and $HP$ intersect at a common point.
2010 Balkan MO Shortlist
Let $c(0, R)$ be a circle with diameter $AB$ and $C$ a point, on it different than $A$ and $B$ such that  $\angle AOC > 90^o$. On the radius $OC$ we consider the point $K$ and the circle $(c_1)$  with center $K$ and radius $KC = R_1$. We draw the tangents $AD$ and $AE$ from $A$ to the circle $(c_1)$. Prove that the straight lines $AC, BK$ and $DE$ are concurrent
2010 Balkan MO Shortlist
Determine whether it is possible to partition $\mathbb{Z}$ into triples $(a,b,c)$ such that, for every triple, $|a^3b + b^3c + c^3a|$ is perfect square.
2010 Balkan MO Shortlist
Solve the following equation in positive integers:

$x^{3} = 2y^{2} + 1 $
2009 Balkan MO Shortlist
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 $\sum_{k=1}^N |x_k|$
2009 Balkan MO Shortlist
Let $ABCD$ be a square and points $M$ $\in$ $BC$, $N \in CD$, $P$ $\in$ $DA$, such that $\angle BAM$ $=$ $x$, $\angle CMN$ $=$ $2x$, $\angle DNP$ $=$ $3x$



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 $\angle APB =4x$
2009 Balkan MO Shortlist
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 $n$.
2009 Balkan MO Shortlist
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*}
2009 Balkan MO Shortlist
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 $\mathbb Z^*$.
2009 Balkan MO Shortlist
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 $\mathbb{Z}[x]$
2009 Balkan MO Shortlist
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*}



Prove that $K_m \geq 0$ for every odd positive integer $m$

Let $M$ $= \prod_{cyc} (x-y)^2$. Prove, $K_7+M^2 K_1 \geq M K_4$
2009 Balkan MO Shortlist
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 $PL^2 = DQ \cdot PQ$.
2009 Balkan MO Shortlist
If $ABCDEF$ is a convex cyclic hexagon, then its diagonals $AD$, $BE$, $CF$ are concurrent if and only if $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$.



Alternative version. Let $ABCDEF$ be a hexagon inscribed in a circle. Then, the lines $AD$, $BE$, $CF$ are concurrent if and only if $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$.
2009 Balkan MO Shortlist
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 $BD$.
2009 Balkan MO Shortlist
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  $(AECZ) = (EBZD) = (ABCD)$.