Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 Balkan MO
In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic.



(Bulgaria)
2013 Balkan MO
Determine all positive integers $x$, $y$ and $z$ such that $x^5 + 4^y = 2013^z$.



(Serbia)
2013 Balkan MO
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:



(a) $xf(x,y,z) = zf(z,y,x)$,



(b) $f(x, ky, k^2z) = kf(x,y,z)$,



(c) $f(1, k, k+1) = k+1$.



(United Kingdom)
2013 Balkan MO
In a mathematical competition, some competitors are friends; friendship is mutual, that is, when $A$ is a friend of $B$, then $B$ is also a friend of $A$.

We say that $n \geq 3$ different competitors $A_1, A_2, \ldots, A_n$ form a weakly-friendly cycle if $A_i$ is not a friend of $A_{i+1}$ for $1 \leq i \leq n$ (where $A_{n+1} = A_1$), and there are no other pairs of non-friends among the components of the cycle.



The following property is satisfied:



"for every competitor $C$ and every weakly-friendly cycle $\mathcal{S}$ of competitors not including $C$, the set of competitors $D$ in $\mathcal{S}$ which are not friends of $C$ has at most one element"



Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.



(Serbia)
2012 Balkan MO
Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$.

Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.
2012 Balkan MO
Prove that

$$\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),$$

for all positive real numbers $x,y$ and $z$.
2012 Balkan MO
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$

Prove that there is a subset $Y$ of $P_n$ such that  $0 \leq y-S_Y < 2^n$
2012 Balkan MO
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:

(i) $f(n!)=f(n)!$ for every positive integer $n$,

(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2010 Balkan MO
Let $a,b$ and $c$ be positive real numbers. Prove that $$ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. $$
2010 Balkan MO
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre  of triangle $PQR$ is the midpoint of the segment $MM_1$.

Prove that $M_1$ lies on the segment $BH_1$.
2010 Balkan MO
A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$.

Prove that $S$ can be covered by a strip of width $2$.
2010 Balkan MO
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$.

Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.
2009 Balkan MO
Solve the equation

$$ 3^x - 5^y = z^2.$$

in positive integers.



Greece
2009 Balkan MO
Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$.



Liubomir Chiriac, Moldova
2009 Balkan MO
A  $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled

i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$,

ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry?



Bulgaria
2009 Balkan MO
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that

$$ f (f^2(m) + 2f^2(n)) = m^2 + 2 n^2$$

for all $ m,n \in S$.



Bulgaria
2008 Balkan MO
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF=PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
2008 Balkan MO
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:

a) $ a_1+a_2+\ldots+a_n\le n^2$

b) $ \frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}\le2008$?
2008 Balkan MO
Let $ n$ be a positive integer. Consider a rectangle $ (90n+1)\times(90n+5)$ consisting of unit squares. Let $ S$ be the set of the vertices of these squares. Prove that the number of distinct lines passing through at least two points of $ S$ is divisible by $ 4$.
2008 Balkan MO
Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1=c$, $ a_{n+1}=a_n^2+a_n+c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2+c^3$ is the $ m$th power of some integer.
2007 Balkan MO
Let $ABCD$ a convex quadrilateral with $AB=BC=CD$, with $AC$ not equal to $BD$ and $E$ be the intersection point of it's diagonals. Prove that $AE=DE$ if and only if $\angle BAD+\angle ADC = 120$.
2007 Balkan MO
Find all real functions $f$ defined on $ \mathbb R$, such that  $$f(f(x)+y) = f(f(x)-y)+4f(x)y ,$$for all real numbers $x,y$.
2007 Balkan MO
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which

$$\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}$$

is a rational number.
2007 Balkan MO
For  a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that

$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.
2006 Balkan MO
Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality

$$ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. $$
2006 Balkan MO
Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$  and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$  and  $C_1D$  pass through the same point.



Greece
2006 Balkan MO
Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers.



Valentin Vornicu, Romania
2006 Balkan MO
Let $m$ be a positive integer and $\{a_n\}_{n\geq 0}$ be a sequence given by $a_0 = a \in \mathbb N$, and $$ a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases}  $$

Find all values of $a$ such that the sequence is periodical (starting from the beginning).
2005 Balkan MO
Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ with the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if and only if $\angle A = 60^\circ$.
2005 Balkan MO
Let $n \geq 2$ be an integer. Let $S$ be a subset of $\{1,2,\dots,n\}$ such that $S$ neither contains two elements one of which divides the other, nor contains two elements which are coprime. What is the maximal possible number of elements of such a set $S$?
2004 Balkan MO
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:

$$ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n})  $$

for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$.
2004 Balkan MO
Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$.
2004 Balkan MO
Let $O$ be an interior point of an acute triangle $ABC$. The circles with centers the midpoints of its sides and passing through $O$ mutually intersect the second time at the points $K$, $L$ and $M$ different from $O$. Prove that $O$ is the incenter of the triangle $KLM$ if and only if $O$ is the circumcenter of the triangle $ABC$.
2004 Balkan MO
The plane is partitioned into regions by a finite number of lines no three of which are concurrent. Two regions are called "neighbors" if the intersection of their boundaries is a segment, or half-line or a line (a point is not a segment). An integer is to be assigned to each region in such a way that:



i) the product of the integers assigned to any two neighbors is less than their sum;

ii) for each of the given lines, and each of the half-planes determined by it, the sum of the integers, assigned to all of the regions lying on this half-plane equal to zero.



Prove that this is possible if and only if not all of the lines are parallel.
2003 Balkan MO
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?
2003 Balkan MO
Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear.



Valentin Vornicu
2003 Balkan MO
Find all functions $f: \mathbb{Q}\to\mathbb{R}$ which fulfill the following conditions:



a) $f(1)+1>0$;



b) $f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy$, for all $x,y\in\mathbb{Q}$;



c) $f(x) = 2f(x+1) +x+2$, for every $x\in\mathbb{Q}$.
2003 Balkan MO
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that $$ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}.  $$
2002 Balkan MO
Consider $n$ points $A_1,A_2,A_3,\ldots, A_n$ ($n\geq 4$) in the plane, such that any three are not collinear. Some pairs of distinct points among $A_1,A_2,A_3,\ldots, A_n$ are connected by segments, such that every point is connected with at least three different points. Prove that there exists $k>1$ and the distinct points $X_1,X_2,\ldots, X_{2k}$ in the set $\{A_1,A_2,A_3,\ldots, A_n\}$, such that for every $i\in \overline{1,2k-1}$ the point $X_i$ is connected with $X_{i+1}$, and $X_{2k}$ is connected with $X_1$.
2002 Balkan MO
Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 = 20$, $ a_2 = 30$ and $ a_{n + 2} = 3a_{n + 1} - a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 + 5a_n a_{n + 1}$ is a perfect square.
2002 Balkan MO
Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.
2002 Balkan MO
Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: $$ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. $$
2001 Balkan MO
Let $a,b,n$ be positive integers such that $2^n - 1 =ab$. Let $k \in \mathbb N$ such that $ab+a-b-1 \equiv 0 \pmod {2^k}$ and $ab+a-b-1 \neq 0 \pmod {2^{k+1}}$. Prove that $k$ is even.
2001 Balkan MO
A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.
2001 Balkan MO
A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?
2000 Balkan MO
Find all functions $f: \mathbb R \to \mathbb R$ such that $$ f( xf(x) + f(y) ) = f^2(x) + y $$ for all $x,y\in \mathbb R$.
2000 Balkan MO
Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$. Let $E$ be a point  on segment $AD$ and $M$ its projection on $BC$. If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.
2000 Balkan MO
How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges?
2000 Balkan MO
Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.
1999 Balkan MO
Let $O$ be the circumcenter of the triangle $ABC$. The segment $XY$ is the diameter of the circumcircle perpendicular to $BC$ and it meets $BC$ at $M$. The point $X$ is closer to $M$ than $Y$ and $Z$ is the point on $MY$ such that $MZ = MX$. The point $W$ is the midpoint of $AZ$.



a) Show that $W$ lies on the circle through the midpoints of the sides of $ABC$;



b) Show that $MW$ is perpendicular to $AY$.
1999 Balkan MO
Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.
1999 Balkan MO
Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to

$AB$, $BC$, $CA$ has area between $\frac 4{27}$ and $\frac 14$.
1999 Balkan MO
Let $\{a_n\}_{n\geq 0}$ be a non-decreasing, unbounded sequence of non-negative integers with $a_0=0$. Let the number of members of the sequence not exceeding $n$ be $b_n$. Prove that $$ (a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1). $$
1998 Balkan MO
Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$, for $k=1,2,\ldots, 1997$. How many distinct terms are there in this sequence?



Greece
1998 Balkan MO
Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality

$$ \sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}.  $$

Bogdan Enescu, Romania
1998 Balkan MO
Let $\mathcal S$ denote the set of points inside or on the border of a triangle $ABC$, without a fixed point $T$ inside the triangle. Show that $\mathcal S$ can be partitioned into disjoint closed segemnts.



Yugoslavia
1998 Balkan MO
Prove that the following equation has no solution in integer numbers: $$ x^2 + 4 = y^5.  $$ Bulgaria
1997 Balkan MO
Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that $$ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , $$ where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$. Prove that $ABCD$ is a square and $O$ is its center.



Yugoslavia
1997 Balkan MO
Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$-element set $A$. If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x$, $y$, prove that $2^k\geq n$.



Yugoslavia
1997 Balkan MO
The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$, and touch a circle $\omega$ internally at $B$ and $C$, respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$. Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$, respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$. Prove that the lines $AD$, $KM$, $LN$ are concurrent.



Greece
1996 Balkan MO
Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively,

prove that $$ OG \leq \sqrt{ R ( R - 2r ) } . $$

Greece
1996 Balkan MO
Let $ p$ be a prime number with $ p>5$. Consider the set $ X = \left\{p - n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$.

Prove that the set $ X$ has two distinct elements $ x$ and $ y$ such that $ x\neq 1$ and $ x\mid y$.



Albania
1996 Balkan MO
In a convex pentagon $ABCDE$, the points $M$, $N$, $P$, $Q$, $R$ are the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EA$, respectively. If the segments $AP$, $BQ$, $CR$ and $DM$ pass through a single point, prove that $EN$ contains that point as well.



Yugoslavia
1996 Balkan MO
Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$, prove that there exist a subset $A$ that satisfies these conditions:



a) $1\in A$ and $2^{1996}-1\in A$;



b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$;



c) The maximum number of elements of $A$ is $2012$.



Romania
1995 Balkan MO
For all real numbers $x,y$ define $x\star y = \frac{ x+y}{ 1+xy}$. Evaluate the expression $$ ( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995. $$



Macedonia
1995 Balkan MO
The circles $\mathcal C_1(O_1, r_1)$ and $\mathcal C_2(O_2, r_2)$, $r_2 > r_1$, intersect at $A$ and $B$ such that $\angle O_1AO_2 = 90^\circ$. The line $O_1O_2$ meets $\mathcal C_1$ at $C$ and $D$, and $\mathcal C_2$ at $E$ and $F$ (in the order $C$, $E$, $D$, $F$). The line $BE$ meets $\mathcal C_1$ at $K$ and $AC$ at $M$, and the line $BD$ meets $\mathcal C_2$ at $L$ and $AF$ at $N$. Prove that

$$ \frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} . $$

Greece
1995 Balkan MO
Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation $$ x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0 $$ are natural numbers, none of which is a perfect square.



Albania
1995 Balkan MO
Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$. Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$. We denote by $a_k$ ($k \geq 0$) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$.



Yugoslavia
1994 Balkan MO
An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$.



Greece
1994 Balkan MO
Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where $$ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n .  $$

Greece
1994 Balkan MO
Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum $$ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . $$

Romania
1994 Balkan MO
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.



Bulgaria
1993 Balkan MO
Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that $$ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. $$ Find the maximum possible value of $f$.



Cyprus
1993 Balkan MO
A positive integer given in decimal representation $\overline{ a_na_{n-1} \ldots a_1a_0 }$ is called monotone if $a_n\leq a_{n-1} \leq \cdots \leq a_0$. Determine the number of monotone positive integers with at most 1993 digits.
1993 Balkan MO
Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$.



Greece
1993 Balkan MO
Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation $$ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m $$has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$.



Romania
1992 Balkan MO
For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$. Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$.



Bulgaria
1992 Balkan MO
Prove that for all positive integers $n$ the following inequality takes place  $$ (2n^2+3n+1)^n \geq 6^n (n!)^2 . $$

Cyprus
1992 Balkan MO
Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is  cyclic, prove that $$ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . $$

Greece
1992 Balkan MO
For each integer $n\geq 3$, find the least natural number $f(n)$ having the property



$\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.
1991 Balkan MO
Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.
1991 Balkan MO
Show that there are infinitely many noncongruent triangles which satisfy the following conditions:

i) the side lengths are relatively prime integers;

ii)the area is an integer number;

iii)the altitudes' lengths are not integer numbers.
1991 Balkan MO
A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$. Show that $P \leq  \frac{3}{2}H$. When does the equality occur?
1991 Balkan MO
Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.
1990 Balkan MO
The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} = 1, a_{2} = 3$, and $ a_{n + 2} = (n + 3)a_{n + 1} - (n + 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.