Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Polish MO Finals
Let $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements  $ A,B$ of the subset there exists a point $ C$ contained in  $ S$ such that the area of triangle $ ABC$ is equal to k .
2009 Polish MO Finals
Let $P,Q,R$  be polynomials of degree at least $1$ with integer coefficients such that for any real number $x$ holds:  $P(Q(x))=Q(R(x))=R(P(x))$. Show that the polynomials $P,Q,R$ are equal.
2009 Polish MO Finals
Let $ x_1,x_2,..,x_n$  be non-negative numbers whose sum is $ 1$  . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$  such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$  and $ 2\leq a_1x_1+a_2x_2+\ldots+a_nx_n\leq 2+\frac{2}{3^n-1}$.
2009 Polish MO Finals
A sphere is inscribed in tetrahedron $ ABCD$ and is tangent to faces  $ BCD,CAD,ABD,ABC$ at points $ P,Q,R,S$ respectively. Segment $ PT$  is the sphere's diameter, and lines $ TA,TQ,TR,TS$ meet the plane $ BCD$ at points $ A',Q',R',S'$. respectively. Show that $ A$  is the center of a circumcircle on the triangle $ S'Q'R'$.
2009 Polish MO Finals
Let $ n$ be a natural number equal or greater than 3 . A sequence of non-negative numbers  $ (c_0,c_1,\ldots,c_n)$ satisfies the condition: $ c_{p}c_{s}+c_{r}c_{t}= c_{p+r}c_{r+s}$ for all non-negative $ p,q,r,s$ such that $ p+q+r+s=n$. Determine all possible values of $ c_2$ when $ c_1=1$.
2008 Polish MO Finals
In each cell of a matrix $ n\times n$ a number from a set $ \{1,2,\ldots,n^2\}$ is written --- in the first row numbers $ 1,2,\ldots,n$, in the second $ n+1,n+2,\ldots,2n$ and so on. Exactly $ n$ of them have been chosen, no two from the same row or the same column. Let us denote by $ a_i$ a number chosen from row number $ i$. Show that:



$$ \frac{1^2}{a_1}+\frac{2^2}{a_2}+\ldots +\frac{n^2}{a_n}\geq \frac{n+2}{2}-\frac{1}{n^2+1}$$
2008 Polish MO Finals
A function $ f: R^3\rightarrow R$ for all reals $ a,b,c,d,e$ satisfies a condition:



$$ f(a,b,c)+f(b,c,d)+f(c,d,e)+f(d,e,a)+f(e,a,b)=a+b+c+d+e$$



Show that for all reals $ x_1,x_2,\ldots,x_n$ ($ n\geq 5$) equality holds:



$$ f(x_1,x_2,x_3)+f(x_2,x_3,x_4)+\ldots +f(x_{n-1},x_n,x_1)+f(x_n,x_1,x_2)=x_1+x_2+\ldots+x_n$$
2008 Polish MO Finals
In a convex pentagon $ ABCDE$ in which $ BC=DE$ following equalities hold:



$$ \angle ABE =\angle CAB =\angle AED-90^{\circ},\qquad \angle ACB=\angle ADE$$



Show that $ BCDE$ is a parallelogram.
2008 Polish MO Finals
Each point of a plane with both coordinates being integers has been colored black or white. Show that there exists an infinite subset of colored points, whose points are in the same color, having a center of symmetry.



[EDIT: added condition about being infinite - now it makes sense]
2008 Polish MO Finals
Let $ R$ be a parallelopiped. Let us assume that areas of all intersections of $ R$ with planes containing centers of three edges of $ R$ pairwisely not parallel and having no common points, are equal. Show that $ R$ is a cuboid.
2008 Polish MO Finals
Let $ S$ be a set of all positive integers which can be represented as $ a^2 + 5b^2$ for some integers $ a,b$ such that $ a\bot b$. Let $ p$ be a prime number such that $ p = 4n + 3$ for some integer $ n$. Show that if for some positive integer $ k$ the number $ kp$ is in $ S$, then $ 2p$ is in $ S$ as well.



Here, the notation $ a\bot b$ means that the integers $ a$ and $ b$ are coprime.
2007 Polish MO Finals
1. In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$. Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$, $L$. Prove that $\frac{LM}{MK}=\frac{AD}{DB}$
2007 Polish MO Finals
2. Positive integer will be called white, if it is equal to $1$ or is a product of even number of primes (not necessarily distinct). Rest of the positive integers will be called black. Determine whether there exists a positive integer which sum of white divisors is equal to sum of black divisors
2007 Polish MO Finals
3. Plane is divided with horizontal and vertical lines into unit squares. Into each square we write a positive integer so that each positive integer appears exactly once. Determine whether it is possible to write numbers in such a way, that each written number is a divisor of a sum of its four neighbours.
2007 Polish MO Finals
4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.
2007 Polish MO Finals
5. In tetrahedron $ABCD$ following equalities hold:

$\angle BAC+\angle BDC=\angle ABD+\angle ACD$

$\angle BAD+\angle BCD=\angle ABC+\angle ADC$

Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.
2007 Polish MO Finals
6. Sequence $a_{0}, a_{1}, a_{2},...$ is determined by $a_{0}=-1$ and

$a_{n}+\frac{a_{n-1}}{2}+\frac{a_{n-2}}{3}+...+\frac{a_{1}}{n}+\frac{a_{0}}{n+1}=0$ for $n\geq 1$

Prove that $a_{n}>0$ for $n\geq 1$
2006 Polish MO Finals
Solve in reals: \begin{eqnarray*}a^2=b^3+c^3 \\ b^2=c^3+d^3 \\ c^2=d^3+e^3 \\ d^2=e^3+a^3 \\ e^2=a^3+b^3 \end{eqnarray*}
2006 Polish MO Finals
Find all positive integers $k$ for which number $3^k+5^k$ is a power of some integer with exponent greater than $1$.
2006 Polish MO Finals
Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
2006 Polish MO Finals
Given a triplet we perform on it the following operation. We choose two numbers among them and change them into their sum and product, left number stays unchanged. Can we, starting from triplet $(3,4,5)$ and performing above operation, obtain again a triplet of numbers which are lengths of right triangle?
2006 Polish MO Finals
Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$. Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.
2006 Polish MO Finals
Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form $$ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0  $$ where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.
2005 Polish MO Finals
Find all triplets $(x,y,n)$ of positive integers which satisfy:

$$ (x-y)^n=xy  $$
2005 Polish MO Finals
The points $A, B, C, D$ lie in this order on a circle $o$.  The point $S$ lies inside $o$ and has properties $\angle SAD=\angle SCB$ and $\angle SDA= \angle SBC$. Line which in which angle bisector of $\angle ASB$ in included cut the circle in points  $P$ and $Q$. Prove $PS =QS$.
2005 Polish MO Finals
In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.
2005 Polish MO Finals
Given real $c > -2$. Prove that for positive reals $x_1,...,x_n$satisfying:$\sum\limits_{i=1}^n \sqrt{x_i ^2+cx_ix_{i+1}+x_{i+1}^2}=\sqrt{c+2}\left( \sum\limits_{i=1}^n x_i \right)$



holds $c=2$ or $x_1=...=x_n$
2005 Polish MO Finals
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by $$x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,$$ has all of its terms relatively prime to $m$.



Proposed by Jaroslaw Wroblewski, Poland
2005 Polish MO Finals
Let  be a convex polygon with $n >5$ vertices and area $1$. Prove  that there exists a convex hexagon inside the given polygon with area at least $\dfrac{3}{4}$
2004 Polish MO Finals
A point $ D$ is taken on the side $ AB$ of a triangle $ ABC$. Two circles passing through $ D$ and touching $ AC$ and $ BC$ at $ A$ and $ B$ respectively intersect again at point $ E$. Let $ F$ be the point symmetric to $ C$ with respect to the perpendicular bisector of $ AB$. Prove that the points $ D,E,F$ lie on a line.
2004 Polish MO Finals
Let $ P$ be a polynomial with integer coefficients such that there are two distinct integers at which $ P$ takes coprime values. Show that there exists an inﬁnite set of integers, such that the values $ P$ takes at them are pairwise coprime.
2004 Polish MO Finals
On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a draw-triple if they can be enumerated so that the ﬁrst defeated the second, the second defeated the third, and the third defeated the ﬁrst. Determine the largest possible number of draw-triples on such a tournament.
2004 Polish MO Finals
Let real numbers $ a,b,c$. Prove that $ \sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\ge \sqrt{3(a+b)^2+3(b+c)^2+3(c+a)^2}$.
2004 Polish MO Finals
Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.
2004 Polish MO Finals
An integer $ m > 1$ is given. The inﬁnite sequence $ (x_n)_{n\ge 0}$ is deﬁned by $ x_i=2^i$ for $ i<m$ and $ x_i=x_{i-1}+x_{i-2}+\cdots +x_{i-m}$ for $ i\ge m$.

Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$.
2003 Polish MO Finals
In an acute-angled triangle $ABC, CD$ is the altitude. A line through the midpoint $M$ of side $AB$ meets the rays $CA$ and $CB$ at $K$ and $L$ respectively such that $CK = CL.$ Point $S$ is the circumcenter of the triangle $CKL.$ Prove that $SD = SM.$
2003 Polish MO Finals
Let $0 < a < 1$ be a real number. Prove that for all finite, strictly increasing sequences $k_1, k_2, \ldots , k_n$ of non-negative integers we have the inequality

$$\biggl( \sum_{i=1}^n a^{k_i} \biggr)^2 < \frac{1+a}{1-a} \sum_{i=1}^n a^{2k_i}.$$
2003 Polish MO Finals
Find all polynomials $W$ with integer coefficients satisfying the following condition: For every natural number $n, 2^n - 1$ is divisible by $W(n).$
2003 Polish MO Finals
A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$
2003 Polish MO Finals
The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.
2003 Polish MO Finals
Let $n$ be an even positive integer. Show that there exists a permutation $(x_1, x_2, \ldots, x_n)$ of the set $\{1, 2, \ldots, n\}$, such that for each $i \in \{1, 2, \ldots, n\}, x_{i+1}$ is one of the numbers $2x_i, 2x_{i}-1, 2x_i - n, 2x_i - n - 1$, where $x_{n+1} = x_1.$
2002 Polish MO Finals
Find all the natural numbers $a,b,c$ such that:



1) $a^2+1$ and $b^2+1$ are primes

2) $(a^2+1)(b^2+1)=(c^2+1)$
2002 Polish MO Finals
On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.
2002 Polish MO Finals
Three non-negative integers are written on a blackboard. A move is to replace two of the integers $k,m$ by $k+m$ and $|k-m|$. Determine whether we can always end with triplet which has at least two zeros
2002 Polish MO Finals
$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: $$ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2},  $$ $$ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} .  $$
2002 Polish MO Finals
There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.
2002 Polish MO Finals
$k$ is a positive integer. The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = k+1$, $a_{n+1} = a_n ^2 - ka_n + k$. Show that $a_m$ and $a_n$ are coprime (for $m \not = n$).
2001 Polish MO Finals
Prove the following inequality:



$x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$



where $\forall _{x_i} x_i > 0$
2001 Polish MO Finals
Given a regular  tetrahedron   $ABCD$   with edge length $1$ and a point $P$ inside it.

What is the maximum value of   $\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|$.
2001 Polish MO Finals
A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$.

Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.
2001 Polish MO Finals
Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.
2001 Polish MO Finals
Let $ABCD$ be a parallelogram and let $K$ and $L$ be points on the segments $BC$ and $CD$, respectively, such that $BK\cdot AD=DL\cdot AB$. Let the lines $DK$ and $BL$ intersect at $P$. Show that $\measuredangle DAP=\measuredangle BAC$.
2001 Polish MO Finals
Given positive integers $n_1<n_2<...<n_{2000}<10^{100}$. Prove that we can choose from the set $\{n_1,...,n_{2000}\}$  nonempty, disjont sets $A$ and $B$ which have the same number of elements, the same sum and the same sum of squares.
2000 Polish MO Finals
Find number of solutions in non-negative reals to the following equations:

\begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}
2000 Polish MO Finals
The sequence $p_1, p_2, p_3, ...$ is defined as follows. $p_1$ and $p_2$ are primes. $p_n$ is the greatest prime divisor of $p_{n-1} + p_{n-2} + 2000$. Show that the sequence is bounded.
2000 Polish MO Finals
$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?
2000 Polish MO Finals
In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even
2000 Polish MO Finals
Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$
1999 Polish MO Finals
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
1999 Polish MO Finals
Given $101$ distinct non-negative integers less than $5050$ show that one can choose four $a, b, c, d$ such that $a + b - c - d$ is a multiple of $5050$
1999 Polish MO Finals
Show that one can find $50$ distinct positive integers such that the sum of each number and its digits is the same.
1999 Polish MO Finals
For which $n$ do the equations have a solution in integers:  \begin{eqnarray*}x_1 ^2 + x_2 ^2 + 50 &=& 16x_1 + 12x_2 \\ x_2 ^2 + x_3 ^2 + 50 &=& 16x_2 + 12x_3 \\ \cdots \quad \cdots \quad \cdots & \cdots & \cdots \quad \cdots \\ x_{n-1} ^2 + x_n ^2 + 50 &=& 16x_{n-1} + 12x_n \\ x_n ^2 + x_1 ^2 + 50 &=& 16x_n + 12x_1 \end{eqnarray*}
1999 Polish MO Finals
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}-a_{j}\right|}+\sum_{i < j}{\left|b_{i}-b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}-b_{j}\right|}$.
1999 Polish MO Finals
Let $ABCDEF$ be a convex hexagon such that $\angle B+\angle D+\angle F=360^{\circ }$ and $$ \frac{AB}{BC} \cdot \frac{CD}{DE} \cdot \frac{EF}{FA} = 1.  $$ Prove that $$ \frac{BC}{CA} \cdot \frac{AE}{EF} \cdot \frac{FD}{DB} = 1.  $$
1998 Polish MO Finals
Find all solutions in positive integers to:

\begin{eqnarray*} a + b + c = xyz \\ x + y + z = abc \end{eqnarray*}
1998 Polish MO Finals
$F_n$ is the Fibonacci sequence $F_0 = F_1 = 1$, $F_{n+2} = F_{n+1} + F_n$. Find all pairs $m > k \geq 0$ such that the sequence $x_0, x_1, x_2, ...$ defined by $x_0 = \frac{F_k}{F_m}$ and $x_{n+1} = \frac{2x_n - 1}{1 - x_n}$ for $x_n \not = 1$, or $1$ if $x_n = 1$, contains the number $1$
1998 Polish MO Finals
$PABCDE$ is a pyramid with $ABCDE$ a convex pentagon. A plane meets the edges $PA, PB, PC, PD, PE$ in points $A', B', C', D', E'$ distinct from $A, B, C, D, E$ and $P$. For each of the quadrilaterals $ABB'A', BCC'B, CDD'C', DEE'D', EAA'E'$ take the intersection of the diagonals. Show that the five intersections are coplanar.
1998 Polish MO Finals
Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?
1998 Polish MO Finals
The points $D, E$ on the side $AB$ of the triangle $ABC$ are such that $\frac{AD}{DB}\frac{AE}{EB} = \left(\frac{AC}{CB}\right)^2$. Show that $\angle ACD = \angle BCE$.
1998 Polish MO Finals
$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?
1997 Polish MO Finals
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.
1997 Polish MO Finals
Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}
1997 Polish MO Finals
In a tetrahedron $ABCD$, the medians of the faces $ABD$, $ACD$, $BCD$ from $D$ make equal angles with the corresponding edges $AB$, $AC$, $BC$. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces.



CommentEquivalent version of the problem: $ABCD$ is a tetrahedron. $DE$, $DF$, $DG$ are medians of triangles $DBC$, $DCA$, $DAB$. The angles between $DE$ and $BC$, between $DF$ and $CA$, and between $DG$ and $AB$ are equal. Show that: area $DBC$ $\leq$ area $DCA$ + area $DAB$.
1997 Polish MO Finals
The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = 0$, $a_n = a_{[n/2]} + (-1)^{n(n+1)/2}$. Show that for any positive integer $k$ we can find $n$ in the range $2^k \leq n < 2^{k+1}$ such that $a_n = 0$.
1997 Polish MO Finals
$ABCDE$ is a convex pentagon such that $DC = DE$ and $\angle C = \angle E = 90^{\cdot}$. $F$ is a point on the side $AB$ such that $\frac{AF}{BF}= \frac{AE}{BC}$. Show that $\angle FCE = \angle ADE$ and $\angle FEC = \angle BDC$.
1997 Polish MO Finals
Given any $n$ points on a unit circle show that at most $\frac{n^2}{3}$ of the segments joining two points have length $> \sqrt{2}$.
1996 Polish MO Finals
Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$.
1996 Polish MO Finals
Let $P$ be a point inside a triangle $ABC$ such that $\angle PBC = \angle PCA < \angle PAB$. The line $PB$ meets the circumcircle of triangle $ABC$ at a point $E$ (apart from $B$). The line $CE$ meets the circumcircle of triangle $APE$ at a point $F$ (apart from $E$). Show that the ratio $\frac{\left|APEF\right|}{\left|ABP\right|}$ does not depend on the point $P$, where the notation $\left|P_1P_2...P_n\right|$ stands for the area of an arbitrary polygon $P_1P_2...P_n$.
1996 Polish MO Finals
$a_i, x_i$ are positive reals such that $a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1$. Show that

$$ 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} $$

When do we have equality?
1996 Polish MO Finals
$ABCD$ is a tetrahedron with $\angle BAC = \angle ACD$ and $\angle ABD = \angle BDC$. Show that $AB = CD$.
1996 Polish MO Finals
Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$
1996 Polish MO Finals
From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition:

$f(i) \geq i-1$ $i=1,...,n$

one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies

$f(i) \leq i+1$ $i=1,...,n$

Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.
1995 Polish MO Finals
How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?
1995 Polish MO Finals
The diagonals of a convex pentagon divide it into a small pentagon and ten triangles. What is the largest number of the triangles that can have the same area?
1995 Polish MO Finals
Let $p$ be a prime number, and define a sequence by: $x_i=i$ for $i=,0,1,2...,p-1$ and $x_n=x_{n-1}+x_{n-p}$ for $n \geq p$

Find the remainder when $x_{p^3}$ is divided by $p$.
1995 Polish MO Finals
The positive reals $x_1, x_2, ... , x_n$ have harmonic mean $1$. Find the smallest possible value of $x_1 + \frac{x_2 ^2}{2} + \frac{x_3 ^3}{3} + ... + \frac{x_n ^n}{n}$.
1995 Polish MO Finals
An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.
1995 Polish MO Finals
$PA, PB, PC$ are three rays in space. Show that there is just one pair of points $B', C$' with $B'$ on the ray $PB$ and $C'$ on the ray $PC$ such that $PC' + B'C' = PA + AB'$ and $PB' + B'C' = PA + AC'$.
1994 Polish MO Finals
Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.
1994 Polish MO Finals
Let be given two parallel lines $k$ and $l$, and a circle not intersecting $k$. Consider a variable point $A$ on the line $k$. The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$. Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$. Prove that all the lines $m$ (as $A$ varies) have a common point.
1994 Polish MO Finals
$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$. Find $\lim_{n \to \infty} \sqrt[n]{a_n}$.
1994 Polish MO Finals
$m, n$ are relatively prime. We have three jugs which contain $m$, $n$ and $m+n$ liters. Initially the largest jug is full of water. Show that for any $k$ in $\{1, 2, ... , m+n\}$ we can get exactly $k$ liters into one of the jugs.
1994 Polish MO Finals
A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$. Show that:

$$ 4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2  $$
1994 Polish MO Finals
The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy:

$$ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd  $$.
1993 Polish MO Finals
Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}
1993 Polish MO Finals
A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.