Navvye/Olympiad-Problems-CSV huggingface dataset:
1996 APMO
The National Marriage Council wishes to invite $n$ couples to form 17 discussion groups under the following conditions:



(1) All members of a group must be of the same sex; i.e. they are either all male or all female.



(2) The difference in the size of any two groups is 0 or 1.



(3) All groups have at least 1 member.



(4) Each person must belong to one and only one group.



Find all values of $n$, $n \leq 1996$, for which this is possible.  Justify your answer.
1996 APMO
Let $a$, $b$, $c$ be the lengths of the sides of a triangle.  Prove that

$$ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c} $$

and determine when equality occurs.
1995 APMO
Determine all sequences of real numbers $a_1$, $a_2$, $\ldots$, $a_{1995}$ which satisfy:

$$ 2\sqrt{a_n - (n - 1)} \geq a_{n+1} - (n - 1), \ \mbox{for} \ n = 1, 2, \ldots 1994,  $$ and $$ 2\sqrt{a_{1995} - 1994} \geq a_1 + 1.  $$
1995 APMO
Let $a_1$, $a_2$, $\ldots$, $a_n$ be a sequence of integers with values between 2 and 1995 such that:

(i) Any two of the $a_i$'s are relatively prime,

(ii) Each $a_i$ is either a prime or a product of primes.

Determine the smallest possible values of $n$ to make sure that the sequence will contain a prime number.
1995 APMO
Let $PQRS$ be a cyclic quadrilateral such that the segments $PQ$ and $RS$ are not parallel.  Consider the set of circles through $P$ and $Q$, and the set of circles through $R$ and $S$.  Determine the set $A$ of points of tangency of circles in these two sets.
1995 APMO
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$.  Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$.  Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
1995 APMO
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
1994 APMO
Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that

(i) For all $x,y \in \Bbb{R}$, $$ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) $$

(ii) For all $x \in [0,1)$, $f(0) \geq f(x)$,

(iii) $-f(-1) = f(1) = 1$.



Find all such functions $f$.
1994 APMO
Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.
1994 APMO
Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$.  Determine all such $n$.
1994 APMO
Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?
1994 APMO
You are given three lists $A$, $B$, and $C$.  List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$.  Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively:

$$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$

Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.
1993 APMO
Let $ABCD$ be a quadrilateral such that all sides have equal length and $\angle{ABC} =60^o$.  Let $l$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$).  Let $E$ and $F$ be the points of intersection of $l$ with $AB$ and $BC$ respectively.  Let $M$ be the point of intersection of $CE$ and $AF$.



Prove that $CA^2 = CM \times CE$.
1993 APMO
Find the total number of different integer values the function $$ f(x) = [x] + [2x] + [\frac{5x}{3}] + [3x] + [4x] $$ takes for real numbers $x$ with $0 \leq x \leq 100$.
1993 APMO
Let

\begin{eqnarray*} f(x) & = & a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \ \ \mbox{and} \\ g(x) & = & c_{n+1} x^{n+1} + c_n x^n + \cdots + c_0 \end{eqnarray*}

be non-zero polynomials with real coefficients such that $g(x) = (x+r)f(x)$ for some real number $r$.  If $a = \max(|a_n|, \ldots, |a_0|)$ and $c = \max(|c_{n+1}|, \ldots, |c_0|)$, prove that $\frac{a}{c} \leq n+1$.
1993 APMO
Determine all positive integers $n$ for which the equation

$$ x^n + (2+x)^n + (2-x)^n = 0 $$

has an integer as a solution.
1993 APMO
Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane

with the following properties:

(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;

(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$.



Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
1992 APMO
A triangle with sides $a$, $b$, and $c$ is given.  Denote by $s$ the semiperimeter, that is $s = \frac{a + b + c}{2}$.  Construct a triangle with sides $s - a$, $s - b$, and $s - c$.  This process is repeated until a triangle can no longer be constructed with the side lengths given.



For which original triangles can this process be repeated indefinitely?
1992 APMO
In a circle $C$ with centre $O$ and radius $r$, let $C_1$, $C_2$ be two circles with centres $O_1$, $O_2$ and radii $r_1$, $r_2$ respectively, so that each circle $C_i$ is internally tangent to $C$ at $A_i$ and so that $C_1$, $C_2$ are externally tangent to each other at $A$.



Prove that the three lines $OA$, $O_1 A_2$, and $O_2 A_1$ are concurrent.
1992 APMO
Let $n$ be an integer such that $n > 3$.  Suppose that we choose three numbers from the set $\{1, 2, \ldots, n\}$.  Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all

possible combinations.

(a) Show that if we choose all three numbers greater than $\frac{n}{2}$, then the values of these combinations are all distinct.

(b) Let $p$ be a prime number such that $p \leq \sqrt{n}$.  Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p - 1$.
1992 APMO
Determine all pairs $(h,s)$ of positive integers with the following property:



If one draws $h$ horizontal lines and another $s$ lines which satisfy

(i) they are not horizontal,

(ii) no two of them are parallel,

(iii) no three of the $h + s$ lines are concurrent,



then the number of regions formed by these $h + s$ lines is 1992.
1992 APMO
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
1991 APMO
Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$.  Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel.  Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$.  Show that triangle $MPQ$ is similar to triangle $ABC$.
1991 APMO
Suppose there are $997$ points given in a plane.  If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane.  Can you find a special case with exactly $1991$ red points?
1991 APMO
Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$.  Show that

$$ \frac{a_1^2}{a_1 + b_1} + \frac{a_2^2}{a_2 + b_2} + \cdots + \frac{a_n^2}{a_n + b_n} \geq \frac{a_1 + a_2 + \cdots + a_n}{2} $$
1991 APMO
During a break, $n$ children at school sit in a circle around their teacher to play a game.  The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule:



He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.



Determine the values of $n$ for which eventually, perhaps after many rounds, all children will have at least one candy each.
1991 APMO
Given are two tangent circles and a point $P$ on their common tangent perpendicular to the lines joining their centres.  Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point $P$.
1990 APMO
Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?
1990 APMO
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time.  Show that

$$ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n $$

for $k = 1$, $2$, $\cdots$, $n - 1$.
1990 APMO
Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$.  For which of these triangles is the product of its altitudes a maximum?
1990 APMO
A set of 1990 persons is divided into non-intersecting subsets in such a way that



1. No one in a subset knows all the others in the subset,



2. Among any three persons in a subset, there are always at least two who do not know each other, and



3. For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them.



(a) Prove that within each subset, every person has the same number of acquaintances.



(b) Determine the maximum possible number of subsets.



Note: It is understood that if a person $A$ knows person $B$, then person $B$ will know person $A$; an acquaintance is someone who is known.  Every person is assumed to know one's self.
1990 APMO
Show that for every integer $n \geq 6$, there exists a convex hexagon which can be dissected into exactly $n$ congruent triangles.
1989 APMO
Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let

$$ S = x_1 + x_2 + \cdots + x_n. $$

Prove that

$$ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} $$
1989 APMO
Prove that the equation $$ 6(6a^2 + 3b^2 + c^2) = 5n^2  $$ has no solutions in integers except $a = b = c = n = 0$.
1989 APMO
Let $A_1$, $A_2$, $A_3$ be three points in the plane, and for convenience, let $A_4= A_1$, $A_5 = A_2$.  For $n = 1$, $2$, and $3$, suppose that $B_n$ is the midpoint of $A_n A_{n+1}$, and suppose that $C_n$ is the midpoint of $A_n B_n$. Suppose that $A_n C_{n+1}$ and $B_n A_{n+2}$ meet at $D_n$, and that $A_n B_{n+1}$ and $C_n A_{n+2}$ meet at $E_n$.

Calculate the ratio of the area of triangle $D_1 D_2 D_3$ to the area of triangle $E_1 E_2 E_3$.
1989 APMO
Let $S$ be a set consisting of $m$ pairs $(a,b)$ of positive integers with the property that $1 \leq a < b \leq n$.  Show that there are at least

$$ 4m \cdot \dfrac{(m - \dfrac{n^2}{4})}{3n} $$

triples $(a,b,c)$ such that $(a,b)$, $(a,c)$, and $(b,c)$ belong to $S$.
1989 APMO
Determine all functions $f$ from the reals to the reals for which



(1) $f(x)$ is strictly increasing and (2) $f(x) + g(x) = 2x$ for all real $x$,



where $g(x)$ is the composition inverse function to $f(x)$.  (Note: $f$ and $g$ are said to be composition inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all real $x$.)
2021 Balkan MO
Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point.
2021 Balkan MO
Find all functions $f:R+$ $\rightarrow$ $R+$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
2021 Balkan MO
Let $a, b$ and $c$ be positive integers satisfying the equation $(a, b) + [a, b]=2021^c$. If $|a-b|$ is a prime number, prove that the number $(a+b)^2+4$ is composite.
2021 Balkan MO
Problem 4. Angel has a warehouse, which initially contains $100$ piles of $100$ pieces of rubbish each. Each morning, Angel performs exactly one of the following moves:

(a) He clears every piece of rubbish from a single pile.

(b) He clears one piece of rubbish from each pile.



However, every evening, a demon sneaks into the warehouse and performs exactly one of the

following moves:

(a) He adds one piece of rubbish to each non-empty pile.

(b) He creates a new pile with one piece of rubbish.



What is the first morning when Angel can guarantee to have cleared all the rubbish from the

warehouse?
2020 Balkan MO
Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$,  and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$.



Proposed by Sam Bealing, United Kingdom
2020 Balkan MO
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,

$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and

$\vspace{0.1cm}$

$\hspace{1cm}ii) f(n)$ divides $n^3$.



Proposed by Dorlir Ahmeti, Albania
2020 Balkan MO
Let $k$ be a positive integer.  Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely:



Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order:

(1) Choose $k+1$ boxes;

(2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add  to the remaining box, if labelled $b_i$, a number of $i$ coins.

(3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.



Proposed by Demetres Christofides, Cyprus
2020 Balkan MO
Let $a_1=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_n$ that has more positive divisors than $a_n$. Prove that $2a_{n+1}=3a_n$ only for finitely many indicies $n$.



 Proposed by Ilija Jovčevski, North Macedonia
2019 Balkan MO
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:

$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$holds for all $p,q\in\mathbb{P}$.



Proposed by Dorlir Ahmeti, Albania
2019 Balkan MO
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$

Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.



(Edit: Proposed by sir Leonard Giugiuc, Romania)
2019 Balkan MO
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:

$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.

$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.

Prove that $KL$ and $ST$ intersect on the line $BC$.
2019 Balkan MO
A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the boundary points. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called boundary lines. Two points in the grid are called neighbours  if the distance between them is equal to $1$.

Anna and Bob play a game on this grid.

Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first.

1) On each of his turns. Bob deletes  at most two boundary points on each boundary line.

2) On each of her turns. Anna makes exactly three steps , where a step  consists of moving her token from its current point to any neighbouring point, which has not been deleted.

As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins.

Does Anna have a winning strategy?



Proposed by Demetres Christofides, Cyprus
2018 Balkan MO
A quadrilateral $ABCD$ is inscribed in a circle $k$ where $AB$ $>$ $CD$,and $AB$ is not paralel to $CD$.Point $M$ is the intersection of diagonals $AC$ and $BD$, and the perpendicular from $M$ to $AB$ intersects the segment $AB$ at a point $E$.If $EM$ bisects the angle $CED$ prove that $AB$ is diameter of $k$.

Proposed by Emil Stoyanov,Bulgaria
2018 Balkan MO
Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$.



Proposed by Jeremy King, UK
2018 Balkan MO
Alice and Bob play the following game: They start with non-empty piles of coins. Taking turns, with Alice playing first, each player choose a pile with an even number of coins and moves half of the coins of this pile to the other pile.  The game ends if a player cannot move, in which case the other player wins.



Determine all pairs $(a,b)$ of positive integers such that if initially the two piles have $a$  and $b$ coins respectively, then Bob has a winning strategy.



Proposed by Dimitris Christophides, Cyprus
2018 Balkan MO
Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$



Proposed by Stanislav Dimitrov,Bulgaria
2017 Balkan MO
Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$
2017 Balkan MO
Consider an acute-angled triangle $ABC$  with $AB<AC$ and let $\omega$ be its circumscribed circle.  Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection.  The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$.  The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$.  The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$.  The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent.



$\text{Vangelis Psychas and Silouanos Brazitikos}$
2017 Balkan MO
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that

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



Proposed by Dorlir Ahmeti, Albania
2017 Balkan MO
On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions:



(A) Gives a candy to the student sitting on his left or to the student sitting on his right.



(B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.



At each step, students perform the actions they have chosen at the same time.



A distribution of candy is called legitimate if it can occur after a finite number of steps.

Find the number of legitimate distributions.



(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)



（Forgive my poor English）
2016 Balkan MO
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$

(Macedonia)
2016 Balkan MO
Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$.



(Greece - Silouanos Brazitikos)
2016 Balkan MO
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.



Note: A monic polynomial has a leading coefficient equal to 1.



(Greece - Panagiotis Lolas and Silouanos Brazitikos)
2016 Balkan MO
The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour.



Note: Any rectangle is assumed here to have sides contained in the lines of the grid.



(Bulgaria - Nikolay Beluhov)
2015 Balkan MO
If ${a, b}$ and $c$ are positive real numbers, prove that



\begin{align*}

 a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.

\end{align*}



(Montenegro).
2015 Balkan MO
Let $\triangle{ABC}$ be a scalene triangle with incentre $I$ and circumcircle $\omega$. Lines $AI, BI, CI$ intersect $\omega$ for the second time at points $D, E, F$, respectively. The parallel lines from $I$ to the sides $BC, AC, AB$ intersect $EF, DF, DE$ at points $K, L, M$, respectively. Prove that the points $K, L, M$ are collinear.

(Cyprus)
2015 Balkan MO
A committee of $3366$ film critics are voting for the Oscars. Every critic voted just an actor and just one actress. After the voting, it was found that for every positive integer $n \in \left \{1, 2, \ldots, 100 \right \}$, there is some actor or some actress who was voted exactly $n$ times. Prove that there are two critics who voted the same actor and the same actress.

(Cyprus)
2015 Balkan MO
Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds



$$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$

where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$.



(Serbia)
2014 Balkan MO
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that $$ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 $$ and determine when equality holds.



UK - David Monk
2014 Balkan MO
A special number is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with $$ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. $$ Prove that



i) there are infinitely many special numbers;

ii) $2014$ is not a special number.



Romania
2014 Balkan MO
Let $ABCD$ be a trapezium inscribed in a circle $\Gamma$ with diameter $AB$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$ . The circle with center $B$ and radius $BE$ meets $\Gamma$ at the points $K$ and $L$ (where $K$ is on the same side of $AB$ as $C$). The line perpendicular to $BD$ at $E$ intersects $CD$ at $M$. Prove that $KM$ is perpendicular to $DL$.



Greece - Silouanos Brazitikos
2014 Balkan MO
Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides.

Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.



UK - Sahl Khan