Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Philippine MO
Find all (if there is one) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$,

$$f(f(x))+xf(x)=1.$$
2011 Philippine MO
The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.

Prove that $4 \leq \chi \leq 7$.
2010 Philippine MO
Find all primes that can be written both as a sum of two primes and as a difference of two primes.
2010 Philippine MO
On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.
2010 Philippine MO
Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.
2010 Philippine MO
There are $2008$ blue, $2009$ red and $2010$ yellow chips on a table. At each step, one chooses two chips of different colors, and recolor both of them using the third color. Can all the chips be of the same color after some steps? Prove your answer.
2010 Philippine MO
Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.
2009 Philippine MO
The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$for every positive integer $n$, where $a_0 = a_1 = 1$. Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$.
2009 Philippine MO
(a) Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$.



(b) Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.
2009 Philippine MO
Each point of a circle is colored either red or blue.



(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.



(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?
2009 Philippine MO
Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$for any positive positive real numbers $a$, $b$ and $c$ with $abc = 1$. Find the minimum value of $k$.
2009 Philippine MO
Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$. Let $E$ be the foot of the perpendicular from $P$ to the line $CD$. Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.
2008 Philippine MO
Prove that the set $\{1, 2, \cdots, 2007\}$ can be expressed as the union of disjoint subsets $A_i$ for $i=1,2,\cdots, 223$ such that each $A_i$ contains nine elements and the sum of all the elements in each $A_i$ is the same.
2008 Philippine MO
Find the largest integer $n$ for which $\frac{n^{2007}+n^{2006}+\cdots+n^2+n+1}{n+2007}$ is an integer.
2008 Philippine MO
Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.
2008 Philippine MO
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{2008^{2x}}{2008+2008^{2x}}$. Prove that

$$\begin{aligned}

f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003.

\end{aligned}$$
2021 Polish MO Finals
Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$  etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$.



exampleFor $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )
2021 Polish MO Finals
Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:



1) $ f(i,i)=0$ for all integers $0\leq i \leq n$



2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$, $j$, $k$, $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$.



Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
2021 Polish MO Finals
Let $\omega$ be the circumcircle of a triangle $ABC$. Let $P$ be any point on $\omega$ different than the verticies of the triangle.

Line $AP$ intersects $BC$ at $D$, $BP$ intersects $AC$ at $E$ and $CP$ intersects $AB$ at $F$.  Let $X$ be the projection of $D$ onto line passing through midpoints of $AP$ and $BC$, $Y$ be the projection of $E$ onto line passing through $BP$ and $AC$ and let $Z$ be the projection of $F$ onto line passing through midpoints of $CP$ and $AB$. Let $Q$ be the circumcenter of triangle $XYZ$. Prove that all possible points $Q$, corresponding to different positions of $P$ lie on one circle.
2021 Polish MO Finals
Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$,

$$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$
2021 Polish MO Finals
A convex hexagon $ABCDEF$ is given where $ \measuredangle FAB + \measuredangle BCD + \measuredangle DEF = 360^{\circ}$ and $ \measuredangle AEB = \measuredangle ADB$. Suppose the lines $AB$ and $DE$ are not parallel. Prove that the circumcenters of the triangles $ \triangle AFE,  \triangle BCD$ and the intersection of the lines $AB$ and $DE$ are collinear.
2021 Polish MO Finals
Given an integer $d \ge 2$ and a circle $\omega$. Hansel drew a finite number of chords of circle $\omega$. The following condition is fulfilled: each end of each chord drawn is at least an end of $d$ different drawn chords. Prove that there is a drawn chord which intersects at least $\tfrac{d^2}{4}$ other drawn chords. Here we assume that the chords with a common end intersect.



Note: Proof that a certain drawn chord crosses at least $\tfrac{d^2}{8}$ other drawn chords will be awarded two points.
2019 Polish MO Finals
Let $ABC$ be an acute triangle. Points $X$ and $Y$ lie on the segments $AB$ and $AC$, respectively, such that $AX=AY$ and the segment $XY$ passes through the orthocenter of the triangle $ABC$. Lines tangent to the circumcircle of the triangle $AXY$ at points $X$ and $Y$ intersect at point $P$. Prove that points $A, B, C, P$ are concyclic.
2019 Polish MO Finals
Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.
2019 Polish MO Finals
$n\ge 3$ guests met at a party. Some of them know each other but there is no quartet of different guests $a, b, c, d$ such that in pairs $\lbrace a, b \rbrace, \lbrace b, c \rbrace, \lbrace c, d \rbrace, \lbrace d, a \rbrace$ guests know each other but in pairs $\lbrace a, c \rbrace, \lbrace b, d \rbrace$ guests don't know each other. We say a nonempty set of guests $X$ is an ingroup, when guests from $X$ know each other pairwise and there are no guests not from $X$ knowing all guests from $X$. Prove that there are at most $\frac{n(n-1)}{2}$ different ingroups at that party.
2019 Polish MO Finals
Let $n, k, \ell$ be positive integers and $\sigma : \lbrace 1, 2, \ldots, n \rbrace \rightarrow \lbrace 1, 2, \ldots, n \rbrace$ an injection such that $\sigma(x)-x\in \lbrace k, -\ell \rbrace$ for all $x\in \lbrace 1, 2, \ldots, n \rbrace$. Prove that $k+\ell|n$.
2019 Polish MO Finals
The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions:

\begin{align*}

\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1

\end{align*}for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that

\begin{align*}

n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}

\end{align*}
2019 Polish MO Finals
Denote by $\Omega$ the circumcircle of the acute triangle $ABC$. Point $D$ is the midpoint of the arc $BC$ of $\Omega$ not containing $A$. Circle $\omega$ centered at $D$ is tangent to the segment $BC$ at point $E$. Tangents to the circle $\omega$ passing through point $A$ intersect line $BC$ at points $K$ and $L$ such that points $B, K, L, C$ lie on the line $BC$ in that order. Circle $\gamma_1$ is tangent to the segments $AL$ and $BL$ and to the circle $\Omega$ at point $M$. Circle $\gamma_2$ is tangent to the segments $AK$ and $CK$ and to the circle $\Omega$ at point $N$. Lines $KN$ and $LM$ intersect at point $P$. Prove that $\sphericalangle KAP = \sphericalangle EAL$.
2018 Polish MO Finals
An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.
2018 Polish MO Finals
A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.
2018 Polish MO Finals
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that

$$f(f(x)+f(y))+cxy=f(x+y).$$
2018 Polish MO Finals
Let $n$ be a positive integer. Suppose there are exactly $M$ squarefree integers $k$ such that $\left\lfloor\frac nk\right\rfloor$ is odd in the set $\{ 1, 2,\ldots, n\}$. Prove $M$ is odd.



An integer is squarefree if it is not divisible by any square other than $1$.
2018 Polish MO Finals
An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.
2018 Polish MO Finals
A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that

$$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$is divisible by $p$. Prove $K+p$ is divisible by $p^2$.
2017 Polish MO Finals
Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.
2017 Polish MO Finals
A sequence $(a_1, a_2,\ldots, a_k)$ consisting of pairwise distinct squares of an $n\times n$ chessboard is called a cycle if $k\geq 4$ and squares $a_i$ and $a_{i+1}$ have a common side for all $i=1,2,\ldots, k$, where $a_{k+1}=a_1$. Subset $X$ of this chessboard's squares is mischievous if each cycle on it contains at least one square in $X$.



Determine all real numbers $C$ with the following property: for each integer $n\geq 2$, on an $n\times n$ chessboard there exists a mischievous subset consisting of at most $Cn^2$ squares.
2017 Polish MO Finals
Integers $a_1, a_2, \ldots, a_n$ satisfy

$$1<a_1<a_2<\ldots < a_n < 2a_1.$$If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$, then prove that

$$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$
2017 Polish MO Finals
Prove that the set of positive integers $\mathbb Z^+$ can be represented as a sum of five pairwise distinct subsets with the following property: each $5$-tuple of numbers of form $(n,2n,3n,4n,5n)$, where $n\in\mathbb Z^+$, contains exactly one number from each of these five subsets.
2017 Polish MO Finals
Point $M$ is the midpoint of $BC$ of a triangle $ABC$, in which $AB=AC$. Point $D$ is the orthogonal projection of $M$ on $AB$. Circle $\omega$ is inscribed in triangle $ACD$ and tangent to segments $AD$ and $AC$ at $K$ and $L$ respectively. Lines tangent to $\omega$ which pass through $M$ cross line $KL$ at $X$ and $Y$, where points $X$, $K$, $L$ and $Y$ lie on $KL$ in this specific order. Prove that points $M$, $D$, $X$ and $Y$ are concyclic.
2017 Polish MO Finals
Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that

$$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$
2016 Polish MO Finals
Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.
2016 Polish MO Finals
Let $ABCD$ be a quadrilateral circumscribed on the circle $\omega$ with center $I$. Assume $\angle BAD+ \angle ADC <\pi$. Let $M, \ N$ be points of tangency of $\omega $ with $AB, \ CD$ respectively. Consider a point $K \in MN$ such that $AK=AM$. Prove that $ID$ bisects the segment $KN$.
2016 Polish MO Finals
Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.
2016 Polish MO Finals
Let $k, n$ be odd positve integers greater than $1$. Prove that if there a exists natural number $a$ such that $k|2^a+1, \ n|2^a-1$, then there is no natural number $b$ satisfying $k|2^b-1, \ n|2^b+1$.
2016 Polish MO Finals
There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions:

$1$. $a< \frac{p}{q} < \frac{r}{s} < b$.

$2.$ $p^2+q^2=r^2+s^2$.
2016 Polish MO Finals
Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.
2015 Polish MO Finals
In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.
2015 Polish MO Finals
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
2015 Polish MO Finals
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers.
2015 Polish MO Finals
Solve the system

$$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$in real numbers.
2015 Polish MO Finals
Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.
2015 Polish MO Finals
Prove that for each positive integer $a$ there exists such an integer $b>a$, for which $1+2^a+3^a$ divides $1+2^b+3^b$.
2014 Polish MO Finals
Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.
2014 Polish MO Finals
Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.
2014 Polish MO Finals
A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.
2014 Polish MO Finals
Denote the set of positive rational numbers by $\mathbb{Q}_{+}$. Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy

$$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$$for all integers $n\ge 1$ and rational numbers $q>0$.
2014 Polish MO Finals
Find all pairs $(x,y)$ of positive integers that satisfy

$$2^x+17=y^4$$.
2014 Polish MO Finals
In an acute triangle $ABC$ point $D$ is the point of intersection of altitude $h_a$ and side $BC$, and points $M, N$ are orthogonal projections of point $D$ on sides $AB$ and $AC$. Lines $MN$ and $AD$ cross the circumcircle of triangle $ABC$ at points $P, Q$ and $A, R$. Prove that point $D$ is the center of the incircle of $PQR$.
2013 Polish MO Finals
Find all solutions of the following equation in integers $x,y:  x^4+ y= x^3+ y^2$
2013 Polish MO Finals
There are given integers $a$ and $b$ such that $a$ is different from $0$ and the number $3+ a +b^2$ is divisible by $6a$. Prove that $a$ is negative.
2013 Polish MO Finals
Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.
2013 Polish MO Finals
Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.
2013 Polish MO Finals
Let k,m and n be three different positive integers. Prove that $$ 

	\left( k-\frac{1}{k} \right)\left( m-\frac{1}{m} \right)\left( n-\frac{1}{n} \right) \le kmn-(k+m+n). $$
2013 Polish MO Finals
For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties:

$1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$

$2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.
2012 Polish MO Finals
Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.
2012 Polish MO Finals
Determine all pairs $(m, n)$ of positive integers, for which cube $K$ with edges of length $n$, can be build in with cuboids of shape $m \times 1 \times 1$ to create cube with edges of length $n + 2$, which has the same center as cube $K$.
2012 Polish MO Finals
Triangle $ABC$ with $AB = AC$ is inscribed in circle $o$. Circles $o_1$ and $o_2$ are internally tangent to circle $o$ in points $P$ and $Q$, respectively, and they are tangent to segments $AB$ and $AC$, respectively, and they are disjoint with the interior of triangle $ABC$. Let $m$ be a line tangent to circles $o_1$ and $o_2$, such that points $P$ and $Q$ lie on the opposite side than point $A$. Line $m$ cuts segments $AB$ and $AC$ in points $K$ and $L$, respectively. Prove, that intersection point of lines $PK$ and $QL$ lies on bisector of angle $BAC$.
2012 Polish MO Finals
$n$ players ($n \ge 4$) took part in the tournament. Each player played exactly one match with every other player, there were no draws. There was no four players $(A, B, C, D)$, such that $A$ won with $B$, $B$ won with $C$, $C$ won with $D$ and $D$ won with $A$. Determine, depending on $n$, maximum number of trios of players $(A, B, C)$, such that  $A$ won with $B$, $B$ won with $C$ and $C$ won with $A$.

(Attention: Trios $(A, B, C)$, $(B, C, A)$ and $(C, A, B)$ are the same trio.)
2012 Polish MO Finals
Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.
2012 Polish MO Finals
Show that for any positive real numbers $a, b, c$ true is inequality:

$\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.
2011 Polish MO Finals
Find all integers $n\geq 1$ such that there exists a permutation $(a_1,a_2,...,a_n)$ of $(1,2,...,n)$ such that $a_1+a_2+...+a_k$ is divisible by $k$ for $k=1,2,...,n$
2011 Polish MO Finals
The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.
2011 Polish MO Finals
Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations

$$\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)\\x_2(x_2+1)=x_3(x_3-1)\\ \vdots \\ x_n(x_n+1) = x_1(x_1-1)\end{array}\right.$$
2011 Polish MO Finals
Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$,

$$f(x)f(y)=g(x)g(y)+g(x)+g(y).$$
2011 Polish MO Finals
In a tetrahedron $ABCD$, the four altitudes are concurrent at $H$. The line $DH$ intersects the plane $ABC$ at $P$ and the circumsphere of $ABCD$ at $Q\neq D$. Prove that $PQ=2HP$.
2011 Polish MO Finals
Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy $$f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.$$
2010 Polish MO Finals
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:

$$a, b, c, a+b, b+c, c+a, a+b+c$$

belong to $S$.
2010 Polish MO Finals
Positive rational number $a$ and $b$ satisfy the equality

$$a^3 + 4a^2b = 4a^2 + b^4.$$

Prove that the number $\sqrt{a}-1$ is a square of a rational number.
2010 Polish MO Finals
$ABCD$ is a parallelogram in which angle $DAB$ is acute. Points $A, P, B, D$ lie on one circle in exactly this order. Lines $AP$ and $CD$ intersect in $Q$. Point $O$ is the circumcenter of the triangle $CPQ$. Prove that if $D \neq O$ then the lines $AD$ and $DO$ are perpendicular.
2010 Polish MO Finals
On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that

$$p_1 > p_2 + 2\cdot \min\{BD, EC\}.$$
2010 Polish MO Finals
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.
2010 Polish MO Finals
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions

$$a_{mn}=a_ma_n, $$ $$a_{m+n} \leq C(a_m + a_n),$$

for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
2009 Polish MO Finals
Each vertex of a convex hexagon is the center of a circle whose radius is equal to the shorter side of the hexagon that contains the vertex. Show that if the intersection of all six circles (including their boundaries) is not empty, then the hexagon is regular.