Navvye/Olympiad-Problems-CSV huggingface dataset:
2017 Czech-Polish-Slovak Match
Each of the ${4n^2}$ unit squares of a ${2n \times 2n}$ board ${(n \ge 1) }$ has been colored blue or red. A set of four different unit squares of the board is called pretty if these squares can be labeled ${A,B,C,D}$ in such a way that ${A}$ and ${B}$ lie in the same row, ${C}$ and ${D}$ lie in the same row, ${A}$ and ${C}$ lie in the same column, ${B}$ and ${D}$ lie in the same column, ${A}$ and ${D}$ are blue, and ${B}$ and ${C}$ are red. Determine the largest possible number of different pretty sets on such a board.



(Poland)
2017 Czech-Polish-Slovak Match
Find all functions ${f : (0, +\infty) \rightarrow R}$ satisfying $f(x) - f(x+ y) = f \left( \frac{x}{y}\right) f(x + y)$ for all $x, y > 0$.



(Austria)
2016 Czech-Polish-Slovak Match
Let $P$ be a non-degenerate polygon with $n$ sides, where $n > 4$. Prove that there exist three distinct vertices $A, B, C$ of $P$ with the following property:If $\ell_1,\ell_2,\ell_3$ are the lengths of the three polygonal chains into which $A, B, C$ break the perimeter of $P$, then there is a triangle with side lengths  $\ell_1,\ell_2$ and $\ell_3$.

Remark: By a non-degenerate polygon we mean a polygon in which every two sides are disjoint, apart from consecutive ones, which share only the common endpoint.(Poland)
2016 Czech-Polish-Slovak Match
Let $m,n > 2$ be even integers. Consider a board of size $m \times n$ whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
2016 Czech-Polish-Slovak Match
Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$.

For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$-balanced.(Czech Republic)
2016 Czech-Polish-Slovak Match
Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system

$(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$

$(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$

$(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$

(Slovakia)
2016 Czech-Polish-Slovak Match
Prove that for every non-negative integer $n$ there exist integers $x, y, z$ with $gcd(x, y, z) = 1$, such that $x^2 + y^2 + z^2 = 3^{2^n}$.(Poland)
2016 Czech-Polish-Slovak Match
Let $ABC$ be an acute-angled triangle with $AB < AC$. Tangent to its circumcircle $\Omega$ at $A$ intersects the line $BC$ at $D$. Let $G$ be the centroid of $\triangle ABC$ and let $AG$ meet $\Omega$ again at $H \neq A$. Suppose the line $DG$ intersects the lines $AB$ and $AC$ at $E$ and $F$, respectively. Prove that $\angle EHG = \angle GHF$.(Slovakia)
2015 Czech-Polish-Slovak Match
On a circle of radius $r$, the distinct points $A$, $B$, $C$, $D$, and $E$ lie in this order, satisfying $AB=CD=DE>r$. Show that the triangle with vertices lying in the centroids of the triangles $ABD$, $BCD$, and $ADE$ is obtuse.



Proposed by Tomáš Jurík, Slovakia
2015 Czech-Polish-Slovak Match
A family of sets $F$ is called perfect if the following condition holds: For every triple of sets $X_1, X_2, X_3\in F$, at least one of the sets $$ (X_1\setminus X_2)\cap X_3,$$ $$(X_2\setminus X_1)\cap X_3$$ is empty. Show that if $F$ is a perfect family consisting of some subsets of a given finite set $U$, then $\left\lvert F\right\rvert\le\left\lvert U\right\rvert+1$.



Proposed by Michał Pilipczuk
2015 Czech-Polish-Slovak Match
Real numbers $x,y,z$ satisfy $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0$$ and none of them lies in the open interval $(-1,1)$. Find the maximum value of $x+y+z$.



Proposed by Jaromír Šimša
2015 Czech-Polish-Slovak Match
A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X\cdot N$ or $X+N$. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results $1\cdot 10=10$, $10+20=30$, $30\cdot 10=300$, and $300+10=310$.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with

(a) $2015$,

(b) $5813$.



Proposed by Michal Rolínek and Peter Novotný
2015 Czech-Polish-Slovak Match
Let $ABC$ be an acute triangle, which is not equilateral. Denote by $O$ and $H$ its circumcenter and orthocenter, respectively. The circle $k$ passes through $B$ and touches the line $AC$ at $A$. The circle $l$ with center on the ray $BH$ touhes the  line $AB$ at $A$. The circles $k$ and $l$ meet in $X$ ($X\ne A$). Show that $\angle HXO=180^\circ-\angle BAC$.



Proposed by Josef Tkadlec
2015 Czech-Polish-Slovak Match
Let $n$ be even positive integer. There are $n$ real positive numbers written on the blackboard. In every step, we choose two numbers, erase them, and replace each of then by their product. Show that for any initial $n$-tuple it is possible to obtain $n$ equal numbers on the blackboard after a finite number of steps.



Proposed by Peter Novotný
2014 Czech-Polish-Slovak Match
Prove that  if the positive real numbers $a, b, c$ satisfy the equation

$a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2)$

then there is an $ABC$ triangle with internal angles $ \alpha, \beta, \gamma$ such that

$ sin \alpha = a$ , $sin \beta = b$  and $sin  \gamma= c$.
2014 Czech-Polish-Slovak Match
For the positive integers $a, b, x_1$ we construct the sequence of numbers $(x_n)_{n=1}^{\infty}$ such that  $x_n = ax_{n-1} + b$ for each $n \ge  2$. Specify the conditions for the given numbers $a, b$ and $x_1$ which are necessary and sufficient for all indexes $m, n$ to apply the implication $m | n \Rightarrow x_m | x_n$.



(Jaromír Šimša)
2014 Czech-Polish-Slovak Match
Given is a convex $ABCD$, which is $ |\angle ABC| = |\angle ADC|= 135^\circ  $. On the $AB, AD$ are also selected points $M, N$ such that  $ |\angle MCD| =   |\angle NCB| = 90^ \circ  $. The circumcircles of the triangles $AMN$ and $ABD$ intersect for the second time at point $K \ne  A$. Prove that lines $AK $ and $KC$ are perpendicular.



(Irán)
2014 Czech-Polish-Slovak Match
Let $ABC$ be a triangle, and let $P$ be the midpoint of $AC$. A circle intersects $AP, CP, BC, AB$ sequentially at their inner points $K, L, M, N$. Let $S$ be the midpoint of   $KL$.  Let also $2 \cdot | AN |\cdot |AB |\cdot |CL | = 2 \cdot | CM | \cdot| BC | \cdot| AK| = | AC | \cdot| AK |\cdot |CL |.$ Prove that if $P\ne S$, then the intersection of $KN$ and $ML$ lies on the perpendicular bisector of the $PS$.



(Jan Mazák)
2014 Czech-Polish-Slovak Match
Let all positive integers $n$ satisfy the following condition:

for each non-negative integers $k, m$ with $k + m \le n$,

the numbers $\binom{n-k}{m}$   and $\binom{n-m}{k}$ leave the same remainder when divided by $2$.



(Poland)





PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak
2014 Czech-Polish-Slovak Match
Let $n \ge 6$ be an integer and $F$ be the system of the $3$-element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition:

for every $1  \le i < j \le  n$ there is at least $ \lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in  F$ such that $i, j \in A$.

Prove that for some integer $m \ge 1$ exist the mutually disjoint  subsets $A_1, A_2 , ... , A_m \in F $ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5 $



(Poland)



PS. just in case my translation does not make sense,

I leave the original in Slovak, in case someone understands something else
2013 Czech-Polish-Slovak Match
Suppose $ABCD$ is a cyclic quadrilateral with $BC = CD$. Let $\omega$ be the circle with center $C$ tangential to the side $BD$. Let $I$ be the centre of the incircle of triangle $ABD$. Prove that the straight line passing through $I$, which is parallel to $AB$, touches the circle $\omega$.
2013 Czech-Polish-Slovak Match
Prove that for every real number $x>0$ and each integer $n>0$ we have

$$x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)$$
2013 Czech-Polish-Slovak Match
For each rational number $r$ consider the statement: If $x$ is a real number such that $x^2-rx$ and $x^3-rx$ are both rational, then $x$ is also rational.



(a) Prove the claim for

$r \ge \frac43$ and $r \le 0$.

(b)  Let $p,q$ be different odd primes such that $3p <4q$. Prove that the claim for $r=\frac{p}q$ does not hold.
2013 Czech-Polish-Slovak Match
Let $a , b$ be integers, and $b$ is not the second power of the whole number.

Prove that  $x^2 + ax + b$ may be the second power of an integer only for finite integer values of $x$



(Martin Panák)
2013 Czech-Polish-Slovak Match
Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle.

[asy]

unitsize(0.25cm);

path p=polygon(3);

for(int m=0; m<=11;++m){

for(int n=0 ; n<= 11-m; ++n){

draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p);

}

}

[/asy]
2013 Czech-Polish-Slovak Match
Let ${ABC}$ be a triangle inscribed in a circle. Point ${P}$ is the center of the  arc ${BAC}$. The circle with the diameter ${CP}$ intersects the angle bisector of angle ${\angle BAC}$  at points ${K, L}$ ${(|AK| <|AL|)}$. Point ${M}$ is the reflection of ${L}$ with respect to line ${BC}$. Prove that the circumcircle of the triangle ${BKM}$ passes through the center of the segment ${BC}$ .
2012 Czech-Polish-Slovak Match
Given a positive integer $n$, let $\tau(n)$ denote the number of positive divisors of $n$ and $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are relatively prime to $n$. Find all $n$ for which one of the three numbers $n,\tau(n), \varphi(n)$ is the arithmetic mean of the other two.
2012 Czech-Polish-Slovak Match
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x+f(y))-f(x)=(x+f(y))^4-x^4$$

for all $x,y \in \mathbb{R}$.
2012 Czech-Polish-Slovak Match
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$. Let $I, J$ and $K$ be the incentres of the triangles $ABC, ACD$ and $ABD$ respectively. Let $E$ be the midpoint of the arc $DB$ of circle $\omega$ containing the point $A$. The line $EK$ intersects again the circle $\omega$ at point $F$ $(F \neq E)$. Prove that the points $C, F, I, J$ lie on a circle.
2012 Czech-Polish-Slovak Match
Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of  the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.
2012 Czech-Polish-Slovak Match
City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$. A street cleaning car starts from point $W$. The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See  too.)

//cdn.artofproblemsolving.com/images/0f190dd2fb1b05de051044bff2c7c2752ff6d252.jpg
2012 Czech-Polish-Slovak Match
Let $a,b,c,d$ be positive real numbers such that

$$abcd=4, \ \ \ \ \ \ \ \ a^2+b^2+c^2+d^2=10$$

Find the maximum possible value of $ab+bc+cd+da$.
2011 Czech-Polish-Slovak Match
Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.
2011 Czech-Polish-Slovak Match
Written on a blackboard are $n$ nonnegative integers whose greatest common divisor is $1$. A move consists of erasing two numbers $x$ and $y$, where $x\ge y$, on the blackboard and replacing them with the numbers $x-y$ and $2y$. Determine for which original $n$-tuples of numbers on the blackboard is it possible to reach a point, after some number of moves, where $n-1$ of the numbers of the blackboard are zeroes.
2011 Czech-Polish-Slovak Match
Points $A$, $B$, $C$, $D$ lie on a circle (in that order) where $AB$ and $CD$ are not parallel. The length of arc $AB$ (which contains the points $D$ and $C$) is twice the length of arc $CD$ (which does not contain the points $A$ and $B$). Let $E$ be a point where $AC=AE$ and $BD=BE$. Prove that if the perpendicular line from point $E$ to the line $AB$ passes through the center of the arc $CD$ (which does not contain the points $A$ and $B$), then $\angle ACB = 108^\circ$.
2011 Czech-Polish-Slovak Match
A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.
2011 Czech-Polish-Slovak Match
In convex quadrilateral $ABCD$, let $M$ and $N$ denote the midpoints of sides $AD$ and $BC$, respectively. On sides $AB$ and $CD$ are points $K$ and $L$, respectively, such that $\angle MKA=\angle NLC$. Prove that if lines $BD$, $KM$, and $LN$ are concurrent, then $$ \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.$$
2011 Czech-Polish-Slovak Match
Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that $$ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a $$ for some integers $n$ and $m$.
2010 Czech-Polish-Slovak Match
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations

$$ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c.  $$
2010 Czech-Polish-Slovak Match
Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.
2010 Czech-Polish-Slovak Match
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
2010 Czech-Polish-Slovak Match
Given any collection of $2010$ nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order:



let $b_1\le b_2\le\cdots\le b_{2011}$ denote the lengths of the blue sides;

let $r_1\le r_2\le\cdots\le r_{2011}$ denote the lengths of the red sides; and

let $w_1\le w_2\le\cdots\le w_{2011}$ denote the lengths of the white sides.



Find the largest integer $k$ for which there necessarily exists at least $k$ indices $j$ such that $b_j$, $r_j$, $w_j$ are the side lengths of a nondegenerate triangle.
2010 Czech-Polish-Slovak Match
Let $x$, $y$, $z$ be positive real numbers satisfying $x+y+z\ge 6$. Find, with proof, the minimum value of $$ x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}. $$
2010 Czech-Polish-Slovak Match
Let $ABCD$ be a convex quadrilateral for which $$ AB+CD=\sqrt{2}\cdot AC\qquad\text{and}\qquad BC+DA=\sqrt{2}\cdot BD.$$ Prove that $ABCD$ is a parallelogram.
2009 Czech-Polish-Slovak Match
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$ \Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1$$ for all $x,y\in\mathbb{R}^+$.
2009 Czech-Polish-Slovak Match
For positive integers $a$ and $k$, define the sequence $a_1,a_2,\ldots$ by $$a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots$$ where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.
2009 Czech-Polish-Slovak Match
Let $\omega$ denote the excircle tangent to side $BC$ of triangle $ABC$. A line $\ell$ parallel to $BC$ meets sides $AB$ and $AC$ at points $D$ and $E$, respectively. Let $\omega'$ denote the incircle of triangle $ADE$. The tangent from $D$ to $\omega$ (different from line $AB$) and the tangent from $E$ to $\omega$ (different from line $AC$) meet at point $P$. The tangent from $B$ to $\omega'$ (different from line $AB$) and the tangent from $C$ to $\omega'$ (different from line $AC$) meet at point $Q$. Prove that, independent of the choice of $\ell$, there is a fixed point that line $PQ$ always passes through.
2009 Czech-Polish-Slovak Match
Given a circle, let $AB$ be a chord that is not a diameter, and let $C$ be a point on the longer arc $AB$. Let $K$ and $L$ denote the reflections of $A$ and $B$, respectively, about lines $BC$ and $AC$, respectively. Prove that the distance between the midpoint of $AB$ and the midpoint of $KL$ is independent of the choice of $C$.
2009 Czech-Polish-Slovak Match
The $n$-tuple $(a_1,a_2,\ldots,a_n)$ of integers satisfies the following:



(i)

$1\le a_1<a_2<\cdots < a_n\le 50$

(ii) for each $n$-tuple $(b_1,b_2,\ldots,b_n)$ of positive integers, there exist a positive integer $m$ and an $n$-tuple $(c_1,c_2,\ldots,c_n)$ of positive integers such that $$mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n. $$ Prove that $n\le 16$ and determine the number of $n$-tuples $(a_1,a_2,\ldots,a_n$) satisfying these conditions for $n=16$.
2009 Czech-Polish-Slovak Match
Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane:  $$ G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.$$ Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.
2008 Czech-Polish-Slovak Match
Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations

$$2x^3=2y(x^2+1)-(z^2+1), $$ $$ 2y^4=3z(y^2+1)-2(x^2+1), $$ $$ 2z^5=4x(z^2+1)-3(y^2+1).$$
2008 Czech-Polish-Slovak Match
$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and  $|AB| =|BC|,$ $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines $AD$, $BE$ and $CF$ are concurrent.
2008 Czech-Polish-Slovak Match
Find all primes $p$ such that the expression

$$\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2$$

is divisible by $p^3$.
2008 Czech-Polish-Slovak Match
Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.
2008 Czech-Polish-Slovak Match
$ABCDE$ is a regular pentagon. Determine the smallest value of the expression

$$\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},$$

where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$.
2008 Czech-Polish-Slovak Match
Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$.
2007 Czech-Polish-Slovak Match
Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$
2007 Czech-Polish-Slovak Match
The Fibonacci sequence is deﬁned by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that  $a_k^4-a_k-2$ is divisible by $m.$
2007 Czech-Polish-Slovak Match
A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter

of $\triangle ABE$ lies on $ED.$
2007 Czech-Polish-Slovak Match
For any real number $p\geq1$ consider the set of all real numbers $x$ with

$$p<x<\left(2+\sqrt{p+\frac{1}{4}}\right)^2.$$

Prove that from any such set one can select four mutually distinct natural numbers $a, b, c, d$ with $ab=cd.$
2007 Czech-Polish-Slovak Match
For which $n\in\{3900, 3901,\cdots, 3909\}$ can the set $\{1, 2, . . . , n\}$ be partitioned into (disjoint) triples in such a way that in each triple one of the numbers equals the sum of the other two?
2007 Czech-Polish-Slovak Match
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $\angle PAB+\angle PDC \leq 90^{\circ}$ and $\angle PBA+\angle PCD \leq 90^{\circ}.$ Prove that $AB+CD\geq BC+AD.$
2006 Czech-Polish-Slovak Match
Five distinct points $A, B, C, D$ and $E$ lie in this order on a circle of radius $r$ and satisfy $AC = BD = CE = r$. Prove that the orthocentres of the triangles $ACD, BCD$ and $BCE$ are the vertices of a right-angled triangle.
2006 Czech-Polish-Slovak Match
There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?
2006 Czech-Polish-Slovak Match
The sum of four real numbers is $9$ and the sum of their squares is $21$. Prove that these numbers can be denoted by $a, b, c, d$ so that $ab-cd \ge 2$ holds.
2006 Czech-Polish-Slovak Match
Show that for every integer $k \ge 1$ there is a positive integer $n$ such that the decimal representation of $2^n$ contains a block of exactly $k$ zeros, i.e. $2^n = \dots a00 \dots 0b \cdots$ with $k$ zeros and $a, b \ne 0$.
2006 Czech-Polish-Slovak Match
Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and

$$a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}$$

for each $n \in \mathbb{N}$.
2006 Czech-Polish-Slovak Match
Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)
2005 Czech-Polish-Slovak Match
Let $n$ be a given positive integer. Solve the system

$$x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,$$

$$x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}$$

in the set of nonnegative real numbers.
2005 Czech-Polish-Slovak Match
A convex quadrilateral $ABCD$ is inscribed in a circle with center $O$ and circumscribed to a circle with center $I$. Its diagonals meet at $P$. Prove that points $O, I$ and $P$ lie on a line.