Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 Mediterranean Mathematics Olympiad
Determine the least integer $k$ for which the following story could hold true:

In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
2013 Mediterranean Mathematics Olympiad
Let $x,y,z$ be positive reals for which:

$\sum (xy)^{2}=6xyz$

Prove that:

$\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.
2013 Mediterranean Mathematics Olympiad
$ABCD$ is quadrilateral inscribed in a circle $\Gamma$ .Lines $AB$ and $CD$ intersect at $E$ and lines$AD$ and $BC$ intersect at $F$.

Prove that the circle with diameter $EF$ and circle $\Gamma$ are orthogonal.
2012 Mediterranean Mathematics Olympiad
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and

$$ \alpha x_n = x_1+x_2+\cdots+x_{n+1}  \mbox{\qquad for } n\ge1. $$ Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.

(Proposed by Gerhard Woeginger, Austria)
2012 Mediterranean Mathematics Olympiad
In an acute $\triangle ABC$, prove that

\begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}
2012 Mediterranean Mathematics Olympiad
Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$.

(Proposed by Gerhard Woeginger, Austria)
2012 Mediterranean Mathematics Olympiad
Let  $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ .

Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$.

Let $k_2$  be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and  $A_2$ denote the respective centers of $k_1$ and $k_2$.

Prove that:

$(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$
2011 Mediterranean Mathematics Olympiad
A Mediterranean polynomial has only real roots and it is of the form

$$ P(x) =  x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 $$ with real coefficients $a_0\ldots,a_7$.  Determine the largest real number that occurs as a root of some Mediterranean polynomial.

(Proposed by Gerhard Woeginger, Austria)
2011 Mediterranean Mathematics Olympiad
Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$.

Show that $|A|\cdot|B|\le|C|^2$.

(Proposed by Gerhard Woeginger, Austria)
2011 Mediterranean Mathematics Olympiad
A regular tetrahedron of height $h$ has a tetrahedron of height $xh$ cut off by a plane parallel to the base. When the remaining frustrum is placed on one of its slant faces on a horizontal plane, it is just on the point of falling over. (In other words, when the remaining frustrum is placed on one of its slant faces on a horizontal plane, the projection of the center of gravity G of the frustrum is a point of the minor base of this slant face.)

Show that $x$ is a root of the equation $x^3 + x^2 + x = 2$.
2011 Mediterranean Mathematics Olympiad
Let $D$ be the foot of the internal bisector of the angle $\angle A$ of the triangle $ABC$. The straight line which joins the incenters of the triangles $ABD$ and $ACD$ cut $AB$ and $AC$ at $M$ and $N$, respectively.

Show that $BN$ and $CM$ meet on the bisector $AD$.
2010 Mediterranean Mathematics Olympiad
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$$$

x^{2}-yz-zu-yu=a$$

$$

y^{2}-zu-ux-xz=b$$

$$

z^{2}-ux-xy-yu=c$$

$$

u^{2}-xy-yz-zx=d$$
2010 Mediterranean Mathematics Olympiad
Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality

$$

\frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}$$





does holds.
2010 Mediterranean Mathematics Olympiad
Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that $$

R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}$$

where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$
2010 Mediterranean Mathematics Olympiad
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$
2009 Mediterranean Mathematics Olympiad
Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled:

- the sum of these real numbers is at least $n$.

- the sum of their squares is at most $4n$.

- the sum of their fourth powers is at least $34n$.

(Proposed by Gerhard Woeginger, Austria)
2009 Mediterranean Mathematics Olympiad
Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively.



Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.
2009 Mediterranean Mathematics Olympiad
Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled:

i) In every row, the entries add up to the same sum $S$.

ii) In every column, the entries also add up to this sum $S$.

iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$.

(Proposed by Gerhard Woeginger, Austria)
2009 Mediterranean Mathematics Olympiad
Let $x,y,z$ be positive real numbers. Prove that

$$ \sum_{cyclic} \frac{xy}{xy+x^2+y^2} ~\le~ \sum_{cyclic} \frac{x}{2x+z} $$

(Proposed by Šefket Arslanagić, Bosnia and Herzegovina)
2008 Mediterranean Mathematics Olympiad
Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if  $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.
2008 Mediterranean Mathematics Olympiad
Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$,

(i)  $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$.

(ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$.

(Proposed by Gerhard Woeginger, Austria)
2008 Mediterranean Mathematics Olympiad
Let $n$ be a positive integer. Calculate the sum $\sum_{k=1}^n\ \  {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}$
2008 Mediterranean Mathematics Olympiad
The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$  for $n>1$.

(a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$.

(b) Find all positive integers  $n$ such that the sum of the roots of polynomial $a_n$ is an integer.
2007 Mediterranean Mathematics Olympiad
Let $x \geq y \geq z$ be real numbers such that $xy + yz + zx = 1$. Prove that $xz < \frac 12.$ Is it possible to improve the value of constant $\frac 12 \ ?$
2007 Mediterranean Mathematics Olympiad
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
2007 Mediterranean Mathematics Olympiad
In the triangle $ABC$, the angle    $\alpha = \angle  BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$
2007 Mediterranean Mathematics Olympiad
Let $x > 1$ be a non-integer number. Prove that

$$\biggl( \frac{x+\{x\}}{[x]} -  \frac{[x]}{x+\{x\}} \biggr) + \biggl( \frac{x+[x]}{ \{x \} } -  \frac{ \{ x \}}{x+[x]} \biggr)  > \frac 92   $$
2006 Mediterranean Mathematics Olympiad
Every point of a plane is colored red or blue, not all with the same color.

Can this be done in such a way that, on every circumference of radius 1,



(a) there is exactly one blue point;

(b) there are exactly two blue points?
2006 Mediterranean Mathematics Olympiad
Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that

$$ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)$$
2006 Mediterranean Mathematics Olympiad
The side lengths $a,b,c$ of a triangle $ABC$ are integers with $\gcd(a,b,c)=1$. The bisector of angle $BAC$ meets $BC$ at $D$.



(a) show that if triangles $DBA$ and $ABC$ are similar then $c$ is a square.



(b) If $c=n^2$ is a square $(n\ge 2)$, find a triangle $ABC$ satisfying (a).
2006 Mediterranean Mathematics Olympiad
Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality



$$ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 $$
2005 Mediterranean Mathematics Olympiad
The professor tells Peter the product of two positive integers and Sam their sum. At first, nobody of them knows the number of the other.



One of them says: You can't possibly guess my number.

Then the other says: You are wrong, the number is 136.



Which number did the professor tell them respectively? Give reasons for your claim.
2005 Mediterranean Mathematics Olympiad
Let $k$ and $k'$ be concentric circles with center $O$ and radius $R$ and $R'$ where $R<R'$ holds. A line passing through $O$ intersects $k$ at $A$ and $k'$ at $B$ where $O$ is between $A$ and $B$. Another line passing through $O$ and distict from $AB$ intersects $k$ at $E$ and $k'$ at $F$ where $E$ is between $O$ and $F$.



Prove that the circumcircles of the triangles $OAE$ and $OBF$, the circle with diameter $EF$ and the circle with diameter $AB$ are concurrent.
2005 Mediterranean Mathematics Olympiad
Let $A_1,A_2,\ldots , A_n$ $(n\geq 3)$ be finite sets of positive integers. Prove, that

$$ \displaystyle \frac{1}{n} \left( \sum_{i=1}^n |A_i|\right) + \frac{1}{{{n}\choose{3}}}\sum_{1\leq i < j < k \leq n} |A_i \cap A_j \cap A_k| \geq \frac{2}{{{n}\choose{2}}}\sum_{1\leq i < j \leq n}|A_i \cap A_j|  $$

holds, where $|E|$ is the cardinality of the set $E$
2005 Mediterranean Mathematics Olympiad
Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$.

Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations
2004 Mediterranean Mathematics Olympiad
Find all natural numbers $m$ such that

$$1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.$$
2004 Mediterranean Mathematics Olympiad
In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that

$$\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .$$
2004 Mediterranean Mathematics Olympiad
Let $a,b,c>0$  and  $ab+bc+ca+2abc=1$  then prove that

$$2(a+b+c)+1\geq 32abc$$
2004 Mediterranean Mathematics Olympiad
Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and

$$\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.$$

If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.
2003 Mediterranean Mathematics Olympiad
Prove that the equation $x^2 + y^2 + z^2 = x + y + z + 1$ has no rational solutions.
2003 Mediterranean Mathematics Olympiad
In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
2003 Mediterranean Mathematics Olympiad
Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality

$$\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.$$
2003 Mediterranean Mathematics Olympiad
Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.
2002 Mediterranean Mathematics Olympiad
Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.
2002 Mediterranean Mathematics Olympiad
Suppose $x, y, a$ are real numbers such that $x+y = x^3 +y^3 = x^5 +y^5 = a$. Find all possible values of $a.$
2002 Mediterranean Mathematics Olympiad
In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$
2002 Mediterranean Mathematics Olympiad
If $a, b, c$ are non-negative real numbers with $ a^2 + b^2 + c^2 = 1$, prove that:

$$ \frac {a}{b^2 + 1} + \frac {b}{c^2 + 1} + \frac {c}{a^2 + 1} \geq \frac {3}{4}(a\sqrt {a} + b\sqrt {b} + c\sqrt {c})^2$$
2001 Mediterranean Mathematics Olympiad
Let $P$ and $Q$ be points on a circle $k$. A chord $AC$ of $k$ passes through the midpoint $M$ of $PQ$. Consider a trapezoid $ABCD$ inscribed in $k$ with $AB \parallel PQ \parallel CD$. Prove that the intersection point $X$ of $AD$ and $BC$ depends only on $k$ and $P,Q.$
2001 Mediterranean Mathematics Olympiad
Find all integers $n$ for which the polynomial $p(x) = x^5 -nx -n -2$ can be represented as a product of two non-constant polynomials with integer coefficients.
2001 Mediterranean Mathematics Olympiad
Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$
2001 Mediterranean Mathematics Olympiad
Let $S$ be the set of points inside a given equilateral triangle $ABC$ with side $1$ or on its boundary. For any $M \in S, a_M, b_M, c_M$ denote the distances from $M$ to $BC,CA,AB$, respectively. Define

$$f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).$$

(a) Describe the set $\{M \in S | f(M) \geq 0\}$ geometrically.



(b) Find the minimum and maximum values of $f(M)$ as well as the points in which these are attained.
2000 Mediterranean Mathematics Olympiad
Let  $F=\{1,2,...,100\}$ and let $G$ be any $10$-element subset of $F$. Prove that there exist two disjoint nonempty subsets $S$ and $T$ of $G$ with the same sum of elements.
2000 Mediterranean Mathematics Olympiad
Suppose that in the exterior of a convex quadrilateral $ABCD$ equilateral triangles $XAB,YBC,ZCD,WDA$ with centroids $S_1,S_2,S_3,S_4$ respectively are constructed. Prove that $S_1S_3\perp S_2S_4$ if and only if $AC=BD$.
2000 Mediterranean Mathematics Olympiad
Let $c_1,c_2,\ldots ,c_n,b_1,b_2,\ldots ,b_n$ $(n\geq 2)$ be positive real numbers. Prove that the equation

$$ \sum_{i=1}^nc_i\sqrt{x_i-b_i}=\frac{1}{2}\sum_{i=1}^nx_i$$

has a unique solution $(x_1,\ldots ,x_n)$ if and only  if $\sum_{i=1}^nc_i^2=\sum_{i=1}^nb_i$.
2000 Mediterranean Mathematics Olympiad
Let $P,Q,R,S$ be the midpoints of the sides $BC,CD,DA,AB$ of a convex quadrilateral, respectively. Prove that

$$4(AP^2+BQ^2+CR^2+DS^2)\le 5(AB^2+BC^2+CD^2+DA^2)$$
1999 Mediterranean Mathematics Olympiad
Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?
1999 Mediterranean Mathematics Olympiad
A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that

this figure can be placed onto a Cartesian plane so that it covers at least $n+1$

points with integer coordinates.
1999 Mediterranean Mathematics Olympiad
Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that $$\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}$$
1999 Mediterranean Mathematics Olympiad
In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that $$ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)$$
1998 Mediterranean Mathematics Olympiad
A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that

$$MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.$$
1998 Mediterranean Mathematics Olympiad
Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.
1998 Mediterranean Mathematics Olympiad
In a triangle $ABC$, $I$ is the incenter and $D,E, F$ are the points of tangency of the incircle with $BC,CA,AB$, respectively. The bisector of angle $BIC$ meets $BC$ at $M$, and the line $AM$ intersects $EF$ at $P$. Prove that $DP$ bisects the angle $FDE$.
2021 Middle European Mathematical Olympiad
Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying

$$ x_{n+1}=A-\frac{1}{x_n} $$for every integer $n \ge 1$, has only finitely many negative terms.
2021 Middle European Mathematical Olympiad
Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots , a_{2r}$ of $2r \ge 4$ pairwise distinct red squares is called a bishop circuit if for every $k \in \{1, \ldots , 2r \}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1}=a_1$ and $a_{2r+2}=a_2$).

In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.

(Remark. Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^\circ$.)
2021 Middle European Mathematical Olympiad
Let $ABC$ be an acute triangle and $D$ an interior point of segment $BC$. Points $E$ and $F$ lie in the half-plane determined by the line $BC$ containing $A$ such that $DE$ is perpendicular to $BE$ and $DE$ is tangent to the circumcircle of $ACD$, while $DF$ is perpendicular to $CF$ and $DF$ is tangent to the circumcircle of $ABD$. Prove that the points $A, D, E$ and $F$ are concyclic.
2021 Middle European Mathematical Olympiad
Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices.

(Remark. For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)
2021 Middle European Mathematical Olympiad
Determine all functions $f: \mathbb{R} \to  \mathbb{R}$ such that the inequality

$$ f(x^2)-f(y^2) \le (f(x)+y)(x-f(y)) $$holds for all real numbers $x$ and $y$.
2021 Middle European Mathematical Olympiad
Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$



there exists an n-pretty polynomial;

any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$.



(Remark. For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)
2021 Middle European Mathematical Olympiad
Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.
2021 Middle European Mathematical Olympiad
Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that:



the diagonals in each triplet intersect in one interior point of the polygon and

all these $n$ intersection points are distinct.
2021 Middle European Mathematical Olympiad
Let $AD$ be the diameter of the circumcircle of an acute triangle $ABC$. The lines through $D$ parallel to $AB$ and $AC$ meet lines $AC$ and $AB$ in points $E$ and $F$, respectively. Lines $EF$ and $BC$ meet at $G$. Prove that $AD$ and $DG$ are perpendicular.
2021 Middle European Mathematical Olympiad
Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P, B, C$, and $Q$ lie in this order on the line $BC$. Prove that $PB=QC$.



Proposed by Dominik Burek, Poland
2021 Middle European Mathematical Olympiad
Find all pairs $(n, p)$ of positive integers such that $p$ is prime and

$$ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). $$
2021 Middle European Mathematical Olympiad
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.
2020 Middle European Mathematical Olympiad
Let $\mathbb{N}$ be the set of positive integers. Determine all positive integers $k$ for which there exist functions $f:\mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N}\to \mathbb{N}$ such that $g$ assumes infinitely many values  and such that $$ f^{g(n)}(n)=f(n)+k$$holds for every positive integer $n$.



(Remark. Here, $f^{i}$ denotes the function $f$ applied $i$ times i.e $f^{i}(j)=f(f(\dots f(j)\dots ))$.)
2020 Middle European Mathematical Olympiad
We call a positive integer $N$ contagious if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers.
2020 Middle European Mathematical Olympiad
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$.

Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.
2020 Middle European Mathematical Olympiad
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$  such that

$$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$