Navvye/Olympiad-Problems-CSV huggingface dataset:
Cono Sur Shortlist - geometry
Let $C_1$ and $C_2$ be two concentric circles and $C_3$ an outer circle to $C_1$ inner to $C_2$ and tangent to both. If the radius of $C_2$ is equal to $ 1$, how much must the radius of $C_1$ be worth, so that the area of is twice that of $C_3$?
Cono Sur Shortlist - geometry
Let $ABCD$ be a quadrilateral and let $O$ be the point of intersection of diagonals $AC$ and $BD$. Knowing that the area of triangle $AOB$ is equal to $ 1$, the area of triangle $BOC$ is equal to $2$, and the area of triangle $COD$ is equal to $4$, calculate the area of triangle $AOD$ and prove that $ABCD$ is a trapezoid.
Cono Sur Shortlist - geometry
Justify the following construction of the bisector of an angle with an inaccessible vertex:



$M \in a$ and $N \in  b$  are taken, the $4$ bisectors of the $4$ internal angles formed by $MN$ are traced with $a$ and $ b$. Said bisectors intersect at $P$ and $Q$, then $PQ$ is the bisector sought.
Cono Sur Shortlist - geometry
Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ?
Cono Sur Shortlist - geometry
A block of houses is a square. There is a courtyard there in which a gold medal has fallen. Whoever calculates how long the side of said apple is, knowing that the distances from the medal to three consecutive corners of the apple are, respectively, $40$ m, $60$ m and $80$ m, will win the medal.
Cono Sur Shortlist - geometry
Consider in the interior of an equilateral triangle $ABC$ points $D, E$ and $F$ such that$ D$ belongs to segment $BE$, $E$ belongs to segment $CF$ and$ F$ to segment $AD$. If $AD=BE = CF$ then $DEF$ is equilateral.
Cono Sur Shortlist - geometry
Let $ABCD$ be a convex quadrilateral, where $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, and $O$ is the intersection of diagonals $AC$ and $BD$. Prove that $O$ is the centroid of the triangle $AMN$ if and only if $ABCD$ is a parallelogram.
Cono Sur Shortlist - geometry
In a triangle $ABC$, let $D$, $E$ and $F$ be the touchpoints of the inscribed circle and the sides $AB$, $BC$ and $CA$. Show that the triangles $DEF$ and $ABC$ are similar if and only if $ABC$ is equilateral.
Cono Sur Shortlist - geometry
Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.
Cono Sur Shortlist - geometry
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$.
Cono Sur Shortlist - geometry
Let $\Gamma$ be a semicircle with center $O$ and diameter $AB$. $D$ is the midpoint  of arc $AB$. On the ray $OD$, we take $E$ such that $OE = BD$. $BE$ intersects the semicircle at $F$ and $ P$ is the point on $AB$ such that $FP$ is perpendicular to $AB$. Prove that $BP=\frac13 AB$.
Cono Sur Shortlist - geometry
Given $4$ lines in the plane such that there are not $2$ parallel to each other or no $3$ concurrent, we consider the following $ 8$ segments: in each line we have $2$ consecutive segments determined by the intersections with the other three lines.

Prove that:

a) The lengths of the $ 8$ segments cannot be the numbers $1, 2, 3,4, 5, 6, 7, 8$ in some order.

b) The lengths of the $ 8$ segments can be $ 8$ different integers.
Cono Sur Shortlist - geometry
Determine the real values of $x$ such that the triangle with sides $5$, $ 8$, and $x$ is obtuse.
Cono Sur Shortlist - geometry
Prove that the sum of the squares of the distances from a point $P$ to the vertices of a triangle $ABC$ is minimum when $ P$ is the centroid of the triangle.
Cono Sur Shortlist - geometry
Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle  CLO$.
Cono Sur Shortlist - geometry
The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$  the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$.  Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.
Cono Sur Shortlist - geometry
An interior $P$ point to a square $ABCD$ is such that $PA = a, PB = b$ and $PC = b + c$, where the numbers $a, b$ and $c$ satisfy the relationship $a^2 = b^2 + c^2$. Prove that the angle $BPC$ is right.
Cono Sur Shortlist - geometry
In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of  triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap  AC$ and $J = HH_2 \cap  H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.
Cono Sur Shortlist - geometry
Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne  H$). Let  $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$  with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.
Cono Sur Shortlist - geometry
Construct triangle given all lenght of it altitudes.





Please, do it elementary with Euclidian geometry (no trigonometry or coordinate geometry).
Cono Sur Shortlist - geometry
Find the ratio between the sum of the areas of the circles and the area of the fourth circle that are shown in the figure

Each circle passes through the center of the previous one and they are internally tangent.
Cono Sur Shortlist - geometry
Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ a point of the minor arc $BC$ of the circle $ABC$ . Let $A_2$ and $A_3$ be points on sides $AB$ and $AC$ respectively such that $\angle BA_1A_2=\angle OAC$ and $\angle CA_1A_3=\angle OAB$ . Points $B_2, B_3, C_2$ and $C_3$ are similarly constructed, with $B_2$ in $BC, B_3$ in $AB, C_2$ in $AC$ and $C_3$ in $BC$. Prove that lines $A_2A_3, B_2B_3$ and $C_2C_3$ are concurrent.
Cono Sur Shortlist - geometry
Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .
Cono Sur Shortlist - geometry
The trapezoid $ABCD$,  of bases $AB$ and $CD$, is inscribed in a circumference $\Gamma$. Let $X$ a variable point of the arc $AB$ of $\Gamma$ that does not contain $C$ or $D$. We denote $Y$ to the point of intersection of $AB$ and $DX$, and let Z be the point of the segment $CX$ such that $\frac{XZ}{XC}=\frac{AY}{AB}$ . Prove that the measure of $\angle AZX$ does not depend on the choice of $X.$
Cono Sur Shortlist - geometry
We have a convex polygon $P$ in the plane and two points $S,T$ in the boundary of $P$, dividing the perimeter in a proportion $1:2$. Three distinct points in the boundary, denoted by $A,B,C$ start to move simultaneously along the boundary, in the same direction and with the same speed. Prove that there will be a moment in which one of the segments $AB, BC, CA$ will have a length smaller or equal than $ST$.
Cono Sur Shortlist - geometry
Let $AA _1$ and $CC_1$ be altitudes of an acute triangle $ABC$. Let $I$ and $J$ be the incenters of the triangles $AA_1C$ and $AC_1C$  respectively. The $C_1J$ and $A_1 I$ lines cut into $T$. Prove that lines  $AT$ and $TC$ are perpendicular.
Cono Sur Shortlist - geometry
Let $ABCD$ be a cyclic quadrilateral. Let $P$ be the intersection of $BC$ and $AD$. Line $AC$ intersects the circumcircle of triangle $BDP$ in points $S$ and $T$, with $S$ between $A$ and $C$. Line $BD$ intersects the circumcircle of triangle $ACP$ in points $U$ and $V$, with $U$ between $B$ and $D$. Prove that $PS$ = $PT$ = $PU$ = $PV$.
Cono Sur Shortlist - geometry
Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.
Cono Sur Shortlist - geometry
Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.
Cono Sur Shortlist - geometry
Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.
Cono Sur Shortlist - geometry
Consider the pentagon $ABCDE$ such that $AB = AE = x$, $AC = AD = y$, $\angle BAE = 90^o$ and $\angle ACB = \angle ADE = 135^o$. It is known that $C$ and $D$ are inside the triangle $BAE$. Determine the length of $CD$ in terms of $x$ and $y$.
Cono Sur Shortlist - geometry
Let $ABC$ be an acute triangle with $AC > AB$. Let $\Gamma$ be the circle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of this circle. Let $I$ be the incenter of $ABC$ and let $E$ and $F$ be points on sides $AB$ and $AC$, respectively, such that $AE = AF$ and $I$ lies on the segment $EF$. Let $P$ be the second intersection point of the circumcircle of the triangle $AEF$ with $\Gamma$ with $P \ne A$. Let $G$ and $H$ be the intersection points of the lines $PE$ and $PF$ with $\Gamma$ different from $P$, respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines AB and $AC$, respectively. Show that the line $JK$ passes through the midpoint of $BC$.
Cono Sur Shortlist - geometry
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$  and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .
Cono Sur Shortlist - geometry
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.
Cono Sur Shortlist - geometry
Let $ABC$ be a triangle whose inscribed circle is $\omega$. Let $r_1$ be the line parallel to $BC$ and tangent to  $\omega$, with $r_1 \ne BC$ and let $r_2$ be the line parallel to $AB$ and tangent to  $\omega$ with $r_2 \ne AB$. Suppose that the intersection point of $r_1$ and $r_2$ lies on the circumscribed circle of triangle $ABC$. Prove that the sidelengths of triangle $ABC$ form an arithmetic progression.
Cono Sur Shortlist - geometry
Let $ABC$ be a triangle with circumcircle $\omega$. The bisector of  $\angle BAC$ intersects $\omega$ at point $A_1$. Let $A_2$ be a point on the segment $AA_1$, $CA_2$ cuts $AB$ and $\omega$ at points $C_1$ and $C_2$, respectively. Similarly, $BA_2$ cuts $AC$ and $\omega$ at points $B_1$ and $B_2$, respectively. Let $M$ be the intersection point of $B_1C_2$  and $B_2C_1$. Prove that $MA_2$ passes the midpoint of $BC$.



proposed by Jhefferson Lopez, Perú
2021 Czech-Polish-Slovak Match
Find all quadruples $(a, b, c, d)$ of positive integers satisfying $\gcd(a, b, c, d) = 1$ and

$$ a | b + c, ~ b | c + d, ~ c | d + a, ~ d | a + b. $$

Vítězslav Kala (Czech Republic)
2021 Czech-Polish-Slovak Match
In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$.



Patrik Bak (Slovakia)
2021 Czech-Polish-Slovak Match
For any two convex polygons $P_1$ and $P_2$ with mutually distinct vertices, denote by $f(P_1, P_2)$ the total number of their vertices that lie on a side of the other polygon. For each positive integer $n \ge 4$, determine

$$ \max \{ f(P_1, P_2) ~ | ~ P_1 ~ \text{and} ~ P_2 ~ \text{are convex} ~ n \text{-gons} \}. $$(We say that a polygon is convex if all its internal angles are strictly less than $180^\circ$.)



Josef Tkadlec (Czech Republic)
2021 Czech-Polish-Slovak Match
Determine the number of $2021$-tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$.



Walther Janous (Austria)
2021 Czech-Polish-Slovak Match
The sequence $a_1, a_2, a_3, \ldots$ satisfies $a_1=1$, and for all $n \ge 2$, it holds that

$$ a_n= \begin{cases}

a_{n-1}+3 ~~ \text{if} ~ n-1 \in \{ a_1,a_2,\ldots,,a_{n-1} \} ; \\
a_{n-1}+2 ~~ \text{otherwise}.

\end{cases} $$Prove that for all positive integers n, we have

$$ a_n < n \cdot (1 + \sqrt{2}). $$

Dominik Burek (Poland) (also known as Burii)
2021 Czech-Polish-Slovak Match
Let $ABC$ be an acute triangle and suppose points $A, A_b, B_a, B, B_c, C_b, C, C_a,$ and $A_c$ lie on its perimeter in this order. Let $A_1 \neq A$ be the second intersection point of the circumcircles of triangles $AA_bC_a$ and $AA_cB_a$. Analogously, $B_1 \neq B$ is the second intersection point of the circumcircles of triangles $BB_cA_b$ and $BB_aC_b$, and $C_1 \neq C$ is the second intersection point of the circumcircles of triangles $CC_aB_c$ and $CC_bA_c$. Suppose that the points $A_1, B_1,$ and $C_1$ are all distinct, lie inside the triangle $ABC$, and do not lie on a single line. Prove that lines $AA_1, BB_1, CC_1,$ and the circumcircle of triangle $A_1B_1C_1$ all pass through a common point.



Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)
2020 Czech-Polish-Slovak Match
Let $ABCD$ be a parallelogram whose diagonals meet at $P$. Denote by $M$ the midpoint of $AB$. Let $Q$ be a point such that $QA$ is tangent to the circumcircle of $MAD$ and $QB$ is tangent to the circumcircle of $MBC$. Prove that points $Q,M,P$ are collinear.



(Patrik Bak, Slovakia)
2020 Czech-Polish-Slovak Match
Given a positive integer $n$,  we say that a real number $x$ is $n$-good if there exist $n$ positive integers $a_1,...,a_n$ such that $$x=\frac{1}{a_1}+...+\frac{1}{a_n}$$Find all positive integers $k$ for which the following assertion is true:

If $a,b$ are real numbers such that the closed interval $[a,b]$ contains infinitely many $2020$-good numbers, then the interval  [a,b] contains at least one $k$-good number.



(Josef Tkadlec, Czech Republic)
2020 Czech-Polish-Slovak Match
The numbers $1, 2,..., 2020$ are written on the blackboard. Venus and Serena play the following game. First, Venus connects by a line segment two numbers such that one of them divides the other. Then Serena connects by a line segment two numbers which has not been connected and such that one of them divides the other. Then Venus again and they continue until there is a triangle with one vertex in $2020$, i.e. $2020$ is connected to two numbers that are connected with each other. The girl that has drawn the last line segment (completed the triangle) is the winner. Which of the girls has a winning strategy?



(Tomáš Bárta, Czech Republic)
2020 Czech-Polish-Slovak Match
Let $a$ be a given real number. Find all functions $f : R \to  R$ such that $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$ holds for all $x,y \in  R$.



(Walther Janous, Austria)
2020 Czech-Polish-Slovak Match
Let $n$ be a positive integer and let $d(n)$ denote the number of ordered pairs of positive integers $(x,y)$ such that

$(x+1)^2-xy(2x-xy+2y)+(y+1)^2=n$. Find the smallest positive integer $n$ satisfying $d(n) = 61$.



(Patrik Bak, Slovakia)
2020 Czech-Polish-Slovak Match
Let $ABC$ be an acute triangle. Let $P$ be a point such that $PB$ and $PC$ are tangent to circumcircle of $ABC$. Let $X$ and $Y$ be variable points on $AB$ and $AC$, respectively, such that $\angle XPY = 2\angle BAC$ and $P$ lies in the interior of triangle $AXY$. Let $Z$ be the reflection of $A$ across $XY$. Prove that the circumcircle of $XYZ$ passes through a fixed point.



(Dominik Burek, Poland)
2019 Czech-Polish-Slovak Match
Let $\omega$ be a circle. Points $A,B,C,X,D,Y$ lie on $\omega$ in this order such that $BD$ is its diameter and $DX=DY=DP$ , where $P$ is the intersection of $AC$ and $BD$. Denote by $E,F$ the intersections of line $XP$ with lines $AB,BC$, respectively. Prove that points $B,E,F,Y$ lie on a single circle.
2019 Czech-Polish-Slovak Match
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$
2019 Czech-Polish-Slovak Match
A dissection of a convex polygon into finitely many triangles by segments is called a trilateration if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is good if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration.
2019 Czech-Polish-Slovak Match
Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$
2019 Czech-Polish-Slovak Match
Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$:



If $a|b$ then $D_a$ is contained in $D_b$.

If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint.



(A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$.)
2019 Czech-Polish-Slovak Match
Let $ABC$ be an acute triangle with $AB<AC$ and $\angle BAC=60^{\circ}$. Denote its altitudes by $AD,BE,CF$ and its orthocenter by $H$. Let $K,L,M$ be the midpoints of sides $BC,CA,AB$, respectively. Prove that the midpoints of segments $AH, DK, EL, FM$ lie on a single circle.
2018 Czech-Polish-Slovak Match
Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$,

$$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$Proposed by Walther Janous, Austria
2018 Czech-Polish-Slovak Match
Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$.



Proposed by Patrik Bak, Slovakia
2018 Czech-Polish-Slovak Match
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.



Proposed by Peter Novotný, Slovakia
2018 Czech-Polish-Slovak Match
Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles.



Proposed by Josef Tkadlec, Czechia
2018 Czech-Polish-Slovak Match
In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points.



Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia
2018 Czech-Polish-Slovak Match
We say that a positive integer $n$ is fantastic if there exist positive rational numbers $a$ and $b$ such that

$$ n = a + \frac 1a + b + \frac 1b.$$(a) Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.

(b) Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.



Proposed by Walther Janous, Austria
2018 Czech-Polish-Slovak Match
Taken from here. Including official solutions in English.
2017 Czech-Polish-Slovak Match
Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer.



(Slovakia)
2017 Czech-Polish-Slovak Match
Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$. Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$. Point ${E}$ lies in the interior of the triangle ${ABC}$, does not lie on the line ${AD}$, and satisfies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$. Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$. Prove that points ${D,E, F,G}$ lie on one circle.



(Slovakia)
2017 Czech-Polish-Slovak Match
Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows.

- In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$.

- Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard.

The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds.



(Poland)
2017 Czech-Polish-Slovak Match
Let ${ABC}$ be a triangle. Line l is parallel to ${BC}$ and it respectively intersects side ${AB}$ at point ${D}$, side ${AC}$ at point ${E}$, and the circumcircle of the triangle ${ABC}$ at points ${F}$ and ${G}$, where points ${F,D,E,G}$ lie in this order on l. The circumcircles of triangles ${FEB}$ and ${DGC}$ intersect at points ${P}$ and ${Q}$. Prove that points ${A, P,Q}$ are collinear.



(Slovakia)