Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 Balkan MO Shortlist
Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of

$$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$taking into consideration all possible choices of triangle $ABC$ and of point $P$.



by Elton Bojaxhiu, Albania
2018 Balkan MO Shortlist
Let $ABC$ be an acute triangle with $AB<AC<BC$  and let $D$ be a point on it's extension of $BC$ towards $C$. Circle $c_1$, with center $A$ and radius $AD$, intersects lines $AC,AB$ and $CB$ at points $E,F$, and $G$ respectively. Circumscribed circle $c_2$ of triangle $AFG$ intersects again lines $FE,BC,GE$ and $DF$ at points $J,H,H' $ and $J'$  respectively. Circumscribed circle $c_3$ of triangle $ADE$ intersects again lines $FE,BC,GE$ and $DF$ at points $I,K,K' $ and $I' $  respectively. Prove that the quadrilaterals $HIJK$ and $H'I'J'K '$ are cyclic and the centers of their circumscribed circles coincide.



by Evangelos Psychas, Greece
2018 Balkan MO Shortlist
In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.
2018 Balkan MO Shortlist
For positive integers $m$ and $n$, let $d(m, n)$ be the number of distinct primes that divide both $m$ and $n$. For instance, $d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2.$ Does there exist a sequence $(a_n)$ of positive integers such that:



$a_1 \geq 2018^{2018};$

$a_m \leq a_n$ whenever $m \leq n$;

$d(m, n) = d(a_m, a_n)$ for all positive integers $m\neq n$?





(Dominic Yeo, United Kingdom)
2018 Balkan MO Shortlist
Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that

$$n!+f(m)!|f(n)!+f(m!)$$for all $m,n\in\mathbb{N}$

Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania
2018 Balkan MO Shortlist
Let $P(x)=a_d x^d+\dots+a_1 x+a_0$ be a non-constant polynomial with non-negative integer coefficients having $d$ rational roots.Prove that $$\text{lcm} \left(P(m),P(m+1),\dots,P(n) \right)\geq m \dbinom{n}{m}$$for all $n>m$



(Navid Safaei, Iran)
2018 Balkan MO Shortlist
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$Prove that $x=1$.



(Silouanos Brazitikos, Greece)
2017 Balkan MO Shortlist
Problem Shortlist BMO 2017







Let $ a $,$ b$,$ c$,  be positive real numbers such that $abc= 1 $. Prove that



$$\frac{1}{a^{5}+b^{5}+c^{2}}+\frac{1}{b^{5}+c^{5}+a^{2}}+\frac{1}{c^{5}+b^{5}+b^{2}}\leq 1 . $$
2017 Balkan MO Shortlist
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$.

Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square.



Proposed by Dorlir Ahmeti, Albania
2017 Balkan MO Shortlist
Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$  with  $a+b+c=1 \}$ and  $f: M\to  R$ given as  $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.
2017 Balkan MO Shortlist
Consider integers $m\ge 2$ and $n\ge 1$.

Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .
2017 Balkan MO Shortlist
A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?
2017 Balkan MO Shortlist
Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.
2017 Balkan MO Shortlist
For any set of points $A_1, A_2,...,A_n$ on the plane, one defines $r( A_1, A_2,...,A_n)$ as the radius of the smallest circle that contains all of these points. Prove that if $n \ge 3$, there are indices $i,j,k$ such that $r( A_1, A_2,...,A_n)=r( A_i, A_j,A_k)$
2017 Balkan MO Shortlist
What is the least positive integer $k$ such that, in every convex $101$-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?
2017 Balkan MO Shortlist
Let $ABC$ be an acute triangle. Variable points $E$ and $F$ are on sides $AC$ and $AB$ respectively such that $BC^2 = BA\cdot  BF + CE \cdot CA$ . As $E$ and $F$ vary prove that the circumcircle of $AEF$ passes through a fixed point other than $A$ .
2017 Balkan MO Shortlist
Let $ABC$ be an acute triangle and $D$ a variable point on side $AC$ . Point $E$ is on $BD$ such that $BE =\frac{BC^2-CD\cdot CA}{BD}$ . As $D$ varies on side $AC$ prove that the circumcircle of $ADE$ passes through a fixed point other than $A$ .
2017 Balkan MO Shortlist
The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.
2017 Balkan MO Shortlist
Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and  $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on  $\omega$.
2017 Balkan MO Shortlist
Construct outside the acute-angled triangle $ABC$ the isosceles triangles $ABA_B, ABB_A , ACA_C,ACC_A ,BCB_C$ and $BCC_B$, so that $$AB = AB_A = BA_B, AC = AC_A=CA_C, BC = BC_B = CB_C$$and $$\angle BAB_A = \angle ABA_B =\angle CAC_A=\angle ACA_C= \angle BCB_C =\angle CBC_B = a < 90^o$$.

Prove that the perpendiculars from $A$ to $B_AC_A$, from $B$ to $A_BC_B$ and from $C$ to $A_CB_C$ are concurrent
2017 Balkan MO Shortlist
Let $ABC$ be an acute triangle with $AB\ne AC$ and circumcircle $\omega$. The angle bisector of $BAC$ intersects $BC$ and $\omega$ at $D$ and $E$ respectively. Circle with diameter $DE$ intersects $\omega$ again at $F \ne E$. Point $P$ is on $AF$ such that $PB = PC$ and $X$ and $Y$ are feet of perpendiculars from $P$ to $AB$ and $AC$ respectively. Let $H$ and $H'$ be the orthocenters of $ABC$ and $AXY$ respectively. $AH$ meets $\omega$ again at $Q$ . If $AH'$ and $HH'$ intersect the circle with diameter $AH$ again at points $S$ and $T$, respectively, prove that the lines $AT , HS$ and $FQ$ are concurrent.
2017 Balkan MO Shortlist
Given an acute triangle  $ABC$ ($AC\ne AB$) and let $(C)$ be its circumcircle. The excircle $(C_1)$ corresponding to the vertex $A$, of center $I_a$, tangents to the side $BC$ at the point $D$ and to the extensions of the sides $AB,AC$ at the points $E,Z$ respectively. Let $I$ and $L$ are the intersection points of the circles $(C)$ and $(C_1)$, $H$  the orthocenter of the triangle $EDZ$ and $N$ the midpoint of segment $EZ$. The parallel line through the point $l_a$ to the line $HL$ meets the line $HI$ at the point $G$. Prove that the perpendicular line $(e)$ through the point $N$ to the line $BC$ and the parallel line $(\delta)$ through the point $G$ to the line $IL$ meet each other on the line $HI_a$.
2017 Balkan MO Shortlist
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
2017 Balkan MO Shortlist
Prove that for all positive integer $n$, there is a positive integer $m$ that $7^n | 3^m +5^m -1$.
2017 Balkan MO Shortlist
Find all pairs of positive integers $(x,y)$ , such that $x^2$ is divisible by $2xy^2 -y^3 +1$.
2017 Balkan MO Shortlist
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts  $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
2016 Balkan MO Shortlist
Let $a, b,c$ be positive real numbers.

Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$
2016 Balkan MO Shortlist
For all $x,y,z>0$ satisfying  $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}\le x+y+z$, prove that

$$\frac{1}{x^2+y+z}+\frac{1}{y^2+z+x}+\frac{1}{z^2+x+y} \le 1$$
2016 Balkan MO Shortlist
The positive real numbers $a, b, c$ satisfy the equality $a + b + c = 1$. For every natural number $n$ find the minimal possible value of the expression $$E=\frac{a^{-n}+b}{1-a}+\frac{b^{-n}+c}{1-b}+\frac{c^{-n}+a}{1-c}$$
2016 Balkan MO Shortlist
Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$.

Find the minimum value and the maximum value of the product $abcd$.
2016 Balkan MO Shortlist
Prove that there is no function from positive real numbers to itself, $f : (0,+\infty)\to(0,+\infty)$ such that:

$f(f(x) + y) = f(x) + 3x + yf(y)$ ,for every $x,y \in (0,+\infty)$



by Greece, Athanasios Kontogeorgis (aka socrates)
2016 Balkan MO Shortlist
Find all integers $n\geq 2$ for which there exist the real numbers $a_k, 1\leq k \leq n$, which are satisfying the following conditions:

$$\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.$$
2016 Balkan MO Shortlist
Find all functions $f : Z \to Z$ for which $f(g(n)) - g(f(n))$ is independent on $n$ for any $g : Z \to Z$.
2016 Balkan MO Shortlist
Let positive integers $K$ and $d$ be given. Prove that there exists a positive integer $n$ and a sequence of $K$ positive integers $b_1,b_2,..., b_K$ such that the number $n$ is a $d$-digit palindrome in all number bases $b_1,b_2,..., b_K$.
2016 Balkan MO Shortlist
There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)

Find the maximal $k$ such that the following holds:

There are $k$ customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.
2016 Balkan MO Shortlist
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.
2016 Balkan MO Shortlist
Given that $ABC$ is a triangle where $AB < AC$. On the half-lines $BA$ and $CA$ we take points $F$ and $E$ respectively such that $BF = CE = BC$. Let $M,N$ and $H$ be the mid-points of the segments $BF,CE$ and $BC$ respectively and $K$ and $O$ be the circumcenters of the triangles $ABC$ and $MNH$ respectively. We assume that $OK$ cuts $BE$ and $HN$ at the points $A_1$ and $B_1$ respectively and that $C_1$ is the point of intersection of $HN$ and $FE$. If the parallel line from $A_1$ to $OC_1$ cuts the line $FE$ at $D$ and the perpendicular from $A_1$ to the line $DB_1$ cuts $FE$ at the point $M_1$, prove that $E$ is the orthocenter of the triangle $A_1OM_1$.
2016 Balkan MO Shortlist
Find all natural numbers $n$ for which $1^{\phi (n)} + 2^{\phi (n)} +... + n^{\phi (n)}$ is coprime with $n$.
2016 Balkan MO Shortlist
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$.

($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).
2016 Balkan MO Shortlist
Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,
2016 Balkan MO Shortlist
A positive integer is called downhill if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial $P(x)$ with rational coefficients takes on an integer value for each downhill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?
2015 Balkan MO Shortlist
Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that:

$$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} 

\leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$

(Albania)
2015 Balkan MO Shortlist
Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that

$$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ 

m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) 

+m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$

(FYROM)
2015 Balkan MO Shortlist
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$

(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)),  \ \ \ \forall x,y\in \mathbb{R}^{+}.$$

(Albania)
2015 Balkan MO Shortlist
Let $m, n$ be positive integers and $a, b$ positive real numbers different from  $1$ such thath $m > n$ and

$$\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c$$. Prove that  $a^m c^n > b^n c^{m}$



(Turkey)
2015 Balkan MO Shortlist
For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid

$$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that  $f(P)=2015$?



(Serbia)
2015 Balkan MO Shortlist
Isaak and Jeremy play the following game.

Isaak says to Jeremy that he thinks a few $2^n$  integers $k_1,..,k_{2^n}$.

Jeremy asks questions of the form: ''Is it true that $k_i<k_j$ ?'' in which Isaak answers by always telling the truth.

After $n2^{n-1}$ questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not.

Prove that Jeremy has bo way to be ''sure'' for his final decision.



(UK)
2015 Balkan MO Shortlist
A chessboard $1000 \times 1000$ is covered by dominoes $1 \times 10$ that can be rotated. We don't know which is the cover, but we are looking for it. For this reason, we choose a few $N$ cells of the chessboard, for which we know the position of the dominoes that cover them.

Which is the minimum $N$ such that after the choice of $N$ and knowing the dominoed that cover them, we can be sure and for the rest of the cover?



(Bulgaria)
2015 Balkan MO Shortlist
In an acute angled triangle $ABC$ , let $BB' $ and  $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle  $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that $$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$(Romania)
2015 Balkan MO Shortlist
Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ του $\omega$ and is different than $B,C$ and the midpoint of arc $BC$.  Tangent of $\Gamma$ on $D$ intersects lines $BC,CA,AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects again the circle  $\omega$ at $F$. Prove that points $D,E,F$ are collinear.



(Saudi Arabia)
2015 Balkan MO Shortlist
A set of points of the plane is called  obtuse-angled if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite  acute-angled set can be extended to an infinite  obtuse-angled set?



(UK)
2015 Balkan MO Shortlist
Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.( Proposed by Emil Stoyanov )
2015 Balkan MO Shortlist
Let $AB$ be a diameter of a circle $(\omega)$  with centre $O$. From an arbitrary point $M$ on $AB$ such that $MA < MB$ we draw the circles $(\omega_1)$ and $(\omega_2)$ with diameters $AM$ and $BM$ respectively. Let $CD$ be an exterior common tangent of $(\omega_1), (\omega_2)$ such that $C$ belongs to $(\omega_1)$ and $D$ belongs to $(\omega_2)$. The point $E$ is diametrically opposite to $C$ with respect to $(\omega_1)$ and the tangent to $(\omega_1)$ at the point $E$ intersects $(\omega_2)$ at the points $F, G$. If the line of the common chord of the circumcircles of the triangles $CED$ and $CFG$ intersects the circle $(\omega)$ at the points $K, L$ and the circle $(\omega_2)$ at the point $N$ (with $N$ closer to $L$), then prove that $KC = NL$.
2015 Balkan MO Shortlist
Let scalene triangle $ABC$ have orthocentre $H$ and circumcircle $\Gamma$. $AH$ meets $\Gamma$ at $D$ distinct from $A$. $BH$ and $CH$ meet $CA$ and $AB$ at $E$ and $F$ respectively, and $EF$ meets $BC$ at $P$. The tangents to $\Gamma$ at $B$ and $C$ meet at $T$. Show that $AP$ and $DT$ are concurrent on the circumcircle of $AFE$.
2015 Balkan MO Shortlist
Let $d$ be an even positive integer.

John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them,  let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$

He continues until two numbers remain written on on the blackboard.

Prove that the sum of squares of those two numbers is different than the numbers  $1^2 ,3^2 ,\ldots,(2n-1)^2$.



(Albania)
2015 Balkan MO Shortlist
Sequence $(a_n)_{n\geq 0}$ is defined as $a_{0}=0, a_1=1, a_2=2, a_3=6$,

and  $ a_{n+4}=2a_{n+3}+a_{n+2}-2a_{n+1}-a_n, n\geq 0$.

Prove that  $n^2$ divides $a_n$ for infinite $n$.



(Romania)
2015 Balkan MO Shortlist
Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $

Let $s,t$ be two different positive integers with the following property:

If $p$ is prime divisor of $s-t$, then  $p$ divides  $a-1$.

Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer.



(FYROM)
2015 Balkan MO Shortlist
Find all pairs of positive integers $(x,y)$ with the following property:

If $a,b$ are relative prime and positive divisors of   $ x^3 + y^3$, then  $a+b - 1$ is divisor of $x^3+y^3$.



(Cyprus)
2015 Balkan MO Shortlist
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$.

Prove that for any positive integer $m$ that:  $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$

(FYROM)
2015 Balkan MO Shortlist
Positive integer $m$ shall be called anagram  of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be anagram  of the sum of the other $3$?



(Bulgaria)
2014 Balkan MO Shortlist
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$
2014 Balkan MO Shortlist
$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$
2014 Balkan MO Shortlist
$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that

$$(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3$$Prove that

$$(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3$$
2014 Balkan MO Shortlist
$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that

$$(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3$$
2014 Balkan MO Shortlist
$\boxed{A6}$The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$
2014 Balkan MO Shortlist
$\boxed{A7}$Prove that for all $x,y,z>0$ with $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ and $0\leq a,b,c<1$ the following inequality holds

$$\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}$$
2014 Balkan MO Shortlist
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?
2014 Balkan MO Shortlist
Let $ M=\{1,2,...,2013\} $ and let $ \Gamma $ be a circle. For every nonempty subset $ B $ of the set $ M $, denote by $ S(B) $ sum of elements of the set $ B $, and define $ S(\varnothing)=0 $ ( $ \varnothing $ is the empty set ). Is it possible to join every subset $ B $ of $ M $ with some point $ A $ on the circle $ \Gamma $ so that following conditions are fulfilled:



$ 1 $. Different subsets are joined with different points;



$ 2 $. All joined points are vertices of a regular polygon;



$ 3 $. If $ A_1,A_2,...,A_k $ are some of the joined points, $ k>2 $ , such that $ A_1A_2...A_k $ is a regular $ k-gon $, then $ 2014 $ divides $ S(B_1)+S(B_2)+...+S(B_k) $ ?
2014 Balkan MO Shortlist
Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$, respectively such that $\angle BAM = \angle MNC$. Suppose that the lines $MN$ and $AB$ intersects at $P$. Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$
2014 Balkan MO Shortlist
Triangle $ABC$ is said to be perpendicular to triangle $DEF$ if the perpendiculars from $A$ to $EF$,from $B$ to $FD$ and from $C$ to $DE$ are concurrent.Prove that if $ABC$ is perpendicular to $DEF$,then $DEF$ is perpendicular to $ABC$
2014 Balkan MO Shortlist
Let $\triangle ABC$ be an isosceles.$(AB=AC)$.Let $D$ and $E$ be two points on the side $BC$ such that $D\in BE$,$E\in DC$ and $2\angle DAE = \angle BAC$.Prove that we can construct a triangle $XYZ$ such that $XY=BD$,$YZ=DE$ and $ZX=EC$.Find $\angle BAC + \angle YXZ$.
2014 Balkan MO Shortlist
Let $A_0B_0C_0$ be a triangle with area equal to $\sqrt 2$. We consider the excenters $A_1$,$B_1$ and $C_1$ then we consider the excenters ,say $A_2,B_2$ and $C_2$,of the triangle $A_1B_1C_1$. By continuing this procedure ,examine if it is possible to arrive to a triangle $A_nB_nC_n$ with all coordinates rational.
2014 Balkan MO Shortlist
Let $ABCD$ be a trapezium inscribed in a circle $k$ with diameter $AB$. A circle with center $B$ and radius $BE$,where $E$ is the intersection point of the diagonals $AC$ and $BD$ meets $k$ at points $K$ and $L$. If the line ,perpendicular to $BD$ at $E$,intersects $CD$ at $M$,prove that $KM\perp DL$.
2014 Balkan MO Shortlist
In $\triangle ABC$ with $AB=AC$,$M$ is the midpoint of $BC$,$H$ is the projection of $M$ onto $AB$ and $D$ is arbitrary point on the side $AC$.Let $E$ be the intersection point of the parallel line through $B$ to $HD$ with the parallel line through $C$ to $AB$.Prove that $DM$ is the bisector of $\angle ADE$.