Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Middle European Mathematical Olympiad
Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds

$$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$

Proposed by Patrik Bak, Slovakia
2019 Middle European Mathematical Olympiad
Let $n\geq 3$ be an integer. We say that a vertex $A_i (1\leq i\leq n)$ of a convex polygon $A_1A_2 \dots A_n$ is Bohemian if its reflection with respect to the midpoint of $A_{i-1}A_{i+1}$ (with $A_0=A_n$ and $A_1=A_{n+1}$) lies inside or on the boundary of the polygon $A_1A_2\dots A_n$. Determine the smallest possible number of Bohemian  vertices a convex $n$-gon can have (depending on $n$).



Proposed by Dominik Burek, Poland
2019 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled triangle with $AC>BC$  and circumcircle $\omega$. Suppose that $P$ is a point on $\omega$ such that $AP=AC$ and that $P$ is an interior point on the shorter arc $BC$ of $\omega$. Let $Q$ be the intersection point of the lines $AP$ and $BC$. Furthermore, suppose that $R$ is a point on $\omega$ such that $QA=QR$ and $R$ is an interior point of the shorter arc $AC$ of $\omega$. Finally, let $S$ be the point of intersection of the line $BC$ with the perpendicular bisector of the side $AB$.

Prove that the points $P, Q, R$ and $S$ are concyclic.



Proposed by Patrik Bak, Slovakia
2019 Middle European Mathematical Olympiad
Determine the smallest positive integer $n$ for which the following statement holds true: From any $n$ consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by $2019$.



Proposed by Kartal Nagy, Hungary
2019 Middle European Mathematical Olympiad
Determine the smallest and the greatest possible values of the expression

$$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.



Proposed by Walther Janous, Austria
2019 Middle European Mathematical Olympiad
Let $\alpha$ be a real number. Determine all polynomials $P$ with real coefficients such that $$P(2x+\alpha)\leq (x^{20}+x^{19})P(x)$$holds for all real numbers $x$.



Proposed by Walther Janous, Austria
2019 Middle European Mathematical Olympiad
There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.

Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).



Proposed by Stephan Wagner, Austria
2019 Middle European Mathematical Olympiad
Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example, to form multidigit numbers (such as $222$) or powers (such as $2^2$).



Valid examples: $$\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \;\; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42$$

Proposed by Stephan Wagner, Austria
2019 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled triangle such that $AB<AC$. Let $D$ be the point of intersection of the perpendicular bisector of the side $BC$ with the side $AC$. Let $P$ be a point on the shorter arc $AC$ of the circumcircle of the triangle $ABC$ such that $DP \parallel BC$. Finally, let $M$ be the midpoint of the side $AB$. Prove that $\angle APD=\angle MPB$.



Proposed by Dominik Burek, Poland
2019 Middle European Mathematical Olympiad
Let $ABC$ be a right-angled triangle with the right angle at $B$ and circumcircle $c$. Denote by $D$ the midpoint of the shorter arc $AB$ of $c$. Let $P$ be the point on the side $AB$ such that $CP=CD$ and let $X$ and $Y$ be two distinct points on $c$ satisfying $AX=AY=PD$. Prove that $X, Y$ and $P$ are collinear.



Proposed by Dominik Burek, Poland
2019 Middle European Mathematical Olympiad
Let $a,b$ and $c$ be positive integers satisfying $a<b<c<a+b$. Prove that $c(a-1)+b$ does not divide $c(b-1)+a$.



Proposed by Dominik Burek, Poland
2019 Middle European Mathematical Olympiad
Let $N$ be a positive integer such that the sum of the squares of all positive divisors of $N$ is equal to the product $N(N+3)$. Prove that there exist two indices $i$ and $j$ such that $N=F_iF_j$ where $(F_i)_{n=1}^{\infty}$ is the Fibonacci sequence defined as $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$.



Proposed by Alain Rossier, Switzerland
2018 Middle European Mathematical Olympiad
Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$  satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$for all $x,y\in Q^+ .$
2018 Middle European Mathematical Olympiad
The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases.

Consider a $2018\times  2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary

cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).
2018 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled  triangle with $AB<AC,$  and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of  triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$  are concurrent.
2018 Middle European Mathematical Olympiad
(a) Prove that for every positive integer $m$ there exists an integer $n\ge m$ such that

$$\left \lfloor \frac{n}{1} \right \rfloor \cdot \left \lfloor \frac{n}{2} \right \rfloor \cdots \left \lfloor \frac{n}{m} \right \rfloor =\binom{n}{m} \\\\\\\\\\\\\\\        (*)$$(b) Denote by $p(m)$ the smallest integer $n \geq m$ such that the equation  $ (*)$ holds. Prove that

$p(2018) = p(2019).$

Remark: For a real number $x,$  we denote by $\left \lfloor x \right \rfloor$  the largest integer not larger than $x.$
2018 Middle European Mathematical Olympiad
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$  Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$
2018 Middle European Mathematical Olympiad
Let $P(x)$ be a polynomial of degree  $n\geq 2$ with rational coefficients such that $P(x) $ has $ n$ pairwise different reel roots forming an arithmetic progression .Prove that among the roots of $P(x) $ there are two that are also the roots of some polynomial of degree $2$ with rational  coefficients .
2018 Middle European Mathematical Olympiad
A graup of pirates had an argument and not each of them holds some other two at gunpoint.All the pirates are called one by one in some order.If the called pirate is still alive , he shoots both pirates  he is aiming at ( some of whom might already be dead .) All shorts are immediatcly  lethal .  After all the pirates have been called , it turns out the exactly $28$ pirates got killed . Prove that if the pirates  were called in whatever other order , at least $10$ pirates would have been killed anyway.
2018 Middle European Mathematical Olympiad
Let $n$ be a positive integer and $u_1,u_2,\cdots ,u_n$ be positive integers not larger than $2^k, $ for some  integer  $k\geq 3.$ A representation of  a non-negative  integer $t$ is a sequence of  non-negative  integers $a_1,a_2,\cdots ,a_n$ such that $t=a_1u_1+a_2u_2+\cdots +a_nu_n.$

Prove that if a non-negative  integer $t$  has a representation,then it also has a  representation  where less than $2k$ of numbers $a_1,a_2,\cdots ,a_n$ are non-zero.
2018 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled  triangle with $AB<AC,$  and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of  $ABC,$  then $DHMO$ is a parallelogram.
2018 Middle European Mathematical Olympiad
Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects  line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$
2018 Middle European Mathematical Olympiad
Let $a_1,a_2,a_3,\cdots$ be the sequence of positive integers such that $$a_1=1

, a_{k+1}=a^3_k+1, $$for all positive integers $k.$

Prove that for every prime number $p$ of the form $3l +2,$ where $l$ is a  non-negative integer ,there exists a  positive integer $n$ such that $a_n$ is divisible by $p.$
2018 Middle European Mathematical Olympiad
An integer $n $ is called silesian if there exist positive integers $a,b$ and $c$ such that $$n=\frac{a^2+b^2+c^2}{ab+bc+ca}.$$$(a)$ prove that there are infinitely many silesian integers.

$(b)$ prove that  not every positive integer is silesian.
2017 Middle European Mathematical Olympiad
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x^2 + f(x)f(y)) = xf(x + y)$$for all real numbers $x$ and $y$.
2017 Middle European Mathematical Olympiad
Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called memorable if the following conditions hold:

(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.

(b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.

Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.
2017 Middle European Mathematical Olympiad
Let $ABCDE$ be a convex pentagon. Let $P$ be the intersection of the lines $CE$ and $BD$. Assume that $\angle PAD = \angle ACB$ and $\angle CAP = \angle EDA$. Prove that the circumcentres of the triangles $ABC$ and $ADE$ are collinear with $P$.
2017 Middle European Mathematical Olympiad
Determine the smallest possible value of

$$|2^m - 181^n|,$$where $m$ and $n$ are positive integers.
2017 Middle European Mathematical Olympiad
Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying

$$P(x + Q(y)) = Q(x + P(y))$$for all real numbers $x$ and $y$.
2017 Middle European Mathematical Olympiad
Determine the smallest possible real constant $C$ such that the inequality

$$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.
2017 Middle European Mathematical Olympiad
There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called bad if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?

(Two lamps are neighbours if their respective cells share a side.)
2017 Middle European Mathematical Olympiad
Let $n \geq 3$ be an integer. A sequence $P_1, P_2, \ldots, P_n$ of distinct points in the plane is called good if no three of them are collinear, the polyline $P_1P_2 \ldots P_n$ is non-self-intersecting and the triangle $P_iP_{i + 1}P_{i + 2}$ is oriented counterclockwise for every $i = 1, 2, \ldots, n - 2$.

For every integer $n \geq 3$ determine the greatest possible integer $k$ with the following property: there exist $n$ distinct points $A_1, A_2, \ldots, A_n$ in the plane for which there are $k$ distinct permutations $\sigma : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}$ such that $A_{\sigma(1)}, A_{\sigma(2)}, \ldots, A_{\sigma(n)}$ is good.

(A polyline $P_1P_2 \ldots P_n$ consists of the segments $P_1P_2, P_2P_3, \ldots, P_{n - 1}P_n$.)
2017 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$. Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$, and let $D$ be the intersection of the rays $AC$ and $BM$. Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$. Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$.

Show that $N$ is the midpoint of the segment $I_B I_C$, where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$, respectively.
2017 Middle European Mathematical Olympiad
Let $ABC$ be an acute-angled triangle with $AB \neq AC$, circumcentre $O$ and circumcircle $\Gamma$. Let the tangents to $\Gamma$ at $B$ and $C$ meet each other at $D$, and let the line $AO$ intersect $BC$ at $E$. Denote the midpoint of $BC$ by $M$ and let $AM$ meet $\Gamma$ again at $N \neq A$. Finally, let $F \neq A$ be a point on $\Gamma$ such that $A, M, E$ and $F$ are concyclic. Prove that $FN$ bisects the segment $MD$.
2017 Middle European Mathematical Olympiad
Determine all integers $n \geq 2$ such that there exists a permutation $x_0, x_1, \ldots, x_{n - 1}$ of the numbers $0, 1, \ldots, n - 1$ with the property that the $n$ numbers

$$x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots  + x_{n - 1}$$are pairwise distinct modulo $n$.
2017 Middle European Mathematical Olympiad
For an integer $n \geq 3$ we define the sequence $\alpha_1, \alpha_2, \ldots, \alpha_k$ as the sequence of exponents in the prime factorization of $n! = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, where $p_1 < p_2 < \ldots < p_k$ are primes. Determine all integers $n \geq 3$ for which $\alpha_1, \alpha_2, \ldots, \alpha_k$ is a geometric progression.
2016 Middle European Mathematical Olympiad
Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which:



(a) $x_j > -1$ for $j = 1, 2,  \ldots, n$ and



(b) $x_1 + x_2 + \ldots + x_n = n.$



Prove that $$ \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} $$and determine when does the equality occur.
2016 Middle European Mathematical Olympiad
There are $n \ge 3$ positive integers written on a board. A move consists of choosing three numbers $a, b, c$ written from the board such that there exists a non-degenerate non-equilateral triangle with sides $a, b, c$ and replacing those numbers with $a + b - c, b + c - a$ and $c + a - b$.



Prove that a sequence of moves cannot be infinite.
2016 Middle European Mathematical Olympiad
Let $ABC$ be an acute triangle such that $\angle BAC > 45^{\circ}$ with circumcenter $O$. A point $P$ is chosen inside triangle $ABC$ such that $A, P, O, B$ are concyclic and the line $BP$ is perpendicular to the line $CP$. A point $Q$ lies on the segment $BP$ such that the line $AQ$ is parallel to the line $PO$.



Prove that $\angle QCB = \angle PCO$.
2016 Middle European Mathematical Olympiad
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.



Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.
2016 Middle European Mathematical Olympiad
Find all triples $(a, b, c)$ of real numbers such that

$$ a^2 + ab + c = 0, $$$$b^2 + bc + a = 0, $$$$c^2 + ca + b = 0.$$
2016 Middle European Mathematical Olympiad
Let $\mathbb{R}$ denote the set of the reals. Find all $f : \mathbb{R} \to \mathbb{R}$ such that

$$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$for all real $x, y$.
2016 Middle European Mathematical Olympiad
A $8 \times 8$ board is given, with sides directed north-south and east-west.

It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one house. A house occupies only one cell.



A house is  in the shade if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.



Find the maximal number of houses that can be placed on the board such that no house is in the shade.
2016 Middle European Mathematical Olympiad
An exam was taken by some students. Each problem was worth $1$ point for the correct answer, and $0$ points for an incorrect one.



For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam.



Prove that there exists a question for which the following holds:

The average score of the students who answered the question correctly is greater than the average score of the students who didn't.
2016 Middle European Mathematical Olympiad
Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$.



Prove that $FP = FQ$.
2016 Middle European Mathematical Olympiad
Let $ABC$ be a triangle for which $AB \neq AC$. Points $K$, $L$, $M$ are the midpoints of the sides $BC$, $CA$, $AB$.

The incircle of $ABC$ with center $I$ is tangent to $BC$ in $D$. A line passing through the midpoint of $ID$ perpendicular to $IK$ meets the line $LM$ in $P$.



Prove that $\angle PIA = 90 ^{\circ}$.
2016 Middle European Mathematical Olympiad
A positive integer $n$ is Mozart if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times.



Prove that:

1. All Mozart numbers are even.

2. There are infinitely many Mozart numbers.
2016 Middle European Mathematical Olympiad
For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers.



Prove that:

1. There does not exist a solution $(a, b, c)$ for $n = 2017$.

2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$.

3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.
2015 Middle European Mathematical Olympiad
Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true:

\begin{align*}

f(a)&=f(b),  \\
 f(a+b)&=\min\{f(a),f(b)\}.

\end{align*}

Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\to Y$ is said to be surjective if for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$.
2015 Middle European Mathematical Olympiad
Let $n\ge 3$ be an integer. An inner diagonal of a simple $n$-gon is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$.



Remark: A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.
2015 Middle European Mathematical Olympiad
Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.
2015 Middle European Mathematical Olympiad
Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that

$$\frac{a^m+b^m}{a^n+b^n}$$

is an integer.
2015 Middle European Mathematical Olympiad
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds:

$$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$
2015 Middle European Mathematical Olympiad
Determine all functions $f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}$ such that

$$f(x^2yf(x))+f(1)=x^2f(x)+f(y)$$

holds for all nonzero real numbers $x$ and $y$.
2015 Middle European Mathematical Olympiad
There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the maximal sum of steps of all students that they can achieve.
2015 Middle European Mathematical Olympiad
Let $N$ be a positive integer. In each of the $N^2$ unit squares of an $N\times N$ board, one of the two diagonals is drawn. The drawn diagonals divide the $N\times N$ board into $K$ regions. For each $N$, determine the smallest and the largest possible values of $K$.



[asy]

draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);

draw((1,0)--(1,3), dotted);

draw((2,0)--(2,3), dotted);

draw((0,1)--(3,1), dotted);

draw((0,2)--(3,2), dotted);

draw((1,0)--(0,1)--(2,3)--(3,2)--(2,1)--(0,3));

draw((1,1)--(2,0)--(3,1));

label("$1$",(0.35,2));

label("$2$",(1,2.65));

label("$3$",(2,2));

label("$4$",(2.65,2.65));

label("$5$",(0.35,0.35));

label("$6$",(1.3,1.3));

label("$7$",(2.65,0.35));

label("Example with $N=3$, $K=7$",(0,-0.3)--(3,-0.3),S);

[/asy]
2015 Middle European Mathematical Olympiad
Let $ABC$ be an acute triangle with $AB>AC$. Prove that there exists a point $D$ with the following property: whenever two distinct points $X$ and $Y$ lie in the interior of $ABC$ such that the points $B$, $C$, $X$, and $Y$ lie on a circle and

$$\angle AXB-\angle ACB=\angle CYA-\angle CBA$$

holds, the line $XY$ passes through $D$.
2015 Middle European Mathematical Olympiad
Let $I$ be the incentre of triangle $ABC$ with $AB>AC$ and let the line $AI$ intersect the side $BC$ at $D$. Suppose that point $P$ lies on the segment $BC$ and satisfies $PI=PD$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$, and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$. Prove that $\angle BAQ=\angle CAJ$.
2015 Middle European Mathematical Olympiad
Find all pairs of positive integers $(a,b)$ such that

$$a!+b!=a^b + b^a.$$
2015 Middle European Mathematical Olympiad
Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.
2014 Middle European Mathematical Olympiad
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that

$$ xf(y) + f(xf(y))  - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)$$

holds for all $x,y \in \mathbb{R}$.
2014 Middle European Mathematical Olympiad
We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A bicoloured triangulation is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive integer $n \ge 4$ triangulable if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater than the number of white triangles of which $A$ is a vertex.



Find all triangulable numbers.
2014 Middle European Mathematical Olympiad
Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$.



Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.