Navvye/Olympiad-Problems-CSV huggingface dataset:
1996 Turkey Team Selection Test
Find the maximum number of pairwise disjoint sets of the form

$S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.
1996 Turkey Team Selection Test
Determine all ordered pairs of positive real numbers $(a, b)$ such that every sequence $(x_{n})$ satisfying $\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0$ must have $\lim_{n \rightarrow \infty} x_n = 0$.
1995 Turkey Team Selection Test
Given real numbers $b \geq a>0$, find all solutions of the system

\begin{align*}

&x_1^2+2ax_1+b^2=x_2,\\
&x_2^2+2ax_2+b^2=x_3,\\
&\qquad\cdots\cdots\cdots\\
&x_n^2+2ax_n+b^2=x_1.

\end{align*}
1995 Turkey Team Selection Test
Let $n$ be a positive integer. Find the number of permutations $\sigma$ of the set $\{1, 2, ..., n\}$ such that $\sigma(j) \geq j$ holds for exactly two values of $j$.
1995 Turkey Team Selection Test
Let $D$ be a point on the small arc $AC$ of the circumcircle of an equilateral triangle $ABC$, different from $A$ and $C$. Let $E$ and $F$ be the projections of $D$ onto $BC$ and $AC$ respectively. Find the locus of the intersection point of $EF$ and $OD$, where $O$ is the center of $ABC$.
1995 Turkey Team Selection Test
In a convex quadrilateral $ABCD$ it is given that $\angle{CAB} = 40^{\circ}, \angle{CAD} = 30^{\circ}, \angle{DBA} = 75^{\circ}$, and $\angle{DBC}=25^{\circ}$. Find $\angle{BDC}$.
1995 Turkey Team Selection Test
Let $n\in\mathbb{N}$ be given. Prove that the following two conditions are equivalent:



$\quad(\text{i})\: n|a^n-a$ for any positive integer $a$;

$\quad(\text{ii})\:$ For any prime divisor $p$ of $n$, $p^2 \nmid n$ and $p-1|n-1$.
1995 Turkey Team Selection Test
The sequence $\{x_n\}$ of real numbers is defined by

$$x_1=1 \quad\text{and}\quad x_{n+1}=x_n+\sqrt[3]{x_n} \quad\text{for}\quad n\geq 1.$$



Show that there exist real numbers $a, b$ such that $\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1$.
1994 Turkey Team Selection Test
$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.
1994 Turkey Team Selection Test
Let $O$ be the center and $[AB]$ be the diameter of a semicircle. $E$ is a point between $O$ and $B$. The perpendicular to $[AB]$ at $E$ meets the semicircle at $D$. A circle which is internally tangent to the arc $\overarc{BD}$ is also tangent to $[DE]$ and $[EB]$ at $K$ and $C$, respectively. Prove that $\widehat{EDC}=\widehat{BDC}$.
1994 Turkey Team Selection Test
All sides and diagonals of a $25$-gon are drawn either red or white. Show that at least $500$ triangles, having all three sides are in same color and having all three vertices from the vertices of the $25$-gon, can be found.
1994 Turkey Team Selection Test
Let $P,Q,R$ be points on the sides of  $\triangle ABC$ such that $P \in [AB],Q\in[BC],R\in[CA]$ and

$\frac{|AP|}{|AB|} = \frac {|BQ|}{|BC|} =\frac{|CR|}{|CA|} =k < \frac 12$

If $G$ is the centroid of $\triangle ABC$, find the ratio $\frac{Area(\triangle PQG)}{Area(\triangle PQR)}$ .
1994 Turkey Team Selection Test
Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that    $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$
1994 Turkey Team Selection Test
Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.
1993 Turkey Team Selection Test
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference.
1993 Turkey Team Selection Test
Let $M$ be the circumcenter of an acute-angled triangle $ABC$. The circumcircle of triangle $BMA$ intersects $BC$ at $P$ and $AC$ at $Q$. Show that $CM \perp PQ$.
1993 Turkey Team Selection Test
Let ($b_n$) be a sequence such that $b_n \geq 0 $ and $b_{n+1}^2 \geq \frac{b_1^2}{1^3}+\cdots+\frac{b_n^2}{n^3}$ for all $n \geq 1$. Prove that there exists a natural number $K$ such that

$$\sum_{n=1}^{K} \frac{b_{n+1}}{b_1+b_2+ \cdots + b_n} \geq \frac{1993}{1000}$$
1993 Turkey Team Selection Test
Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that



$(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them;

$(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.
1993 Turkey Team Selection Test
Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle.  Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent.
1993 Turkey Team Selection Test
Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy:

$$f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}$$
1992 Turkey Team Selection Test
Is there $14$ consecutive positive integers  such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.
1992 Turkey Team Selection Test
The line passing through $B$ is perpendicular to the side $AC$ at $E$. This line meets the circumcircle of $\triangle ABC$ at $D$. The foot of the perpendicular from $D$ to the side $BC$ is $F$. If $O$ is the center of the circumcircle of $\triangle ABC$, prove that $BO$ is perpendicular to $EF$.
1992 Turkey Team Selection Test
$x_1, x_2,\cdots,x_{n+1}$  are posive real numbers satisfying the equation

$\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$



Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.
1992 Turkey Team Selection Test
The feet of perpendiculars from the intersection point of the diagonals of cyclic quadrilateral $ABCD$ to the sides $AB,BC,CD,DA$ are $P,Q,R,S$, respectively. Prove $PQ+RS=QR+SP$.
1992 Turkey Team Selection Test
There are $n$ boxes which is numbere from $1$ to $n$. The box with number $1$ is open, and the others are closed. There are  $m$ identical balls ($m\geq n$). One of the balls is put into the open box, then we open the box with number $2$. Now, we put another ball to one of two open boxes, then we open the box with number $3$. Go on until the last box will be open. After that the remaining balls will be randomly put into the boxes.  In how many ways this arrangement can be done?
1992 Turkey Team Selection Test
A circle with radius $4$ and $251$ distinct points inside the circle are given. Show that it is possible to draw a circle with radius $1$ and containing at least $11$ of these points.
1991 Turkey Team Selection Test
Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.
1991 Turkey Team Selection Test
Show that the equation $a^2+b^2+c^2+d^2=a^2\cdot b^2\cdot c^2\cdot d^2$ has no solution in positive integers.
1991 Turkey Team Selection Test
Let $f$ be a function on defined on $|x|<1$ such that  $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i  $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$  where $p(x)$ and $q(x)$ are polynomials with integer coefficients.
1991 Turkey Team Selection Test
A frog is jumping on $N$ stones which are numbered from $1$  to $N$ from left to right. The frog is jumping to the previous stone (to the left) with probability  $p$ and is jumping to the next stone (to the right) with probability $1-p$. If the frog has jumped to the left from the leftmost stone or to the right from the rightmost stone, it will fall into the water. The frog is initially on the leftmost stone. If $p< \tfrac 13$, show that the frog will fall into the water from the rightmost stone with a probability higher than  $\tfrac 12$.
1991 Turkey Team Selection Test
$p$  passengers get on a train with $n$ wagons. Find the probability of being at least one passenger at each wagon.
1991 Turkey Team Selection Test
Let $U$ be the sum of lengths of sides of a tetrahedron (triangular pyramid) with vertices $O,A,B,C$. Let $V$ be the volume of the convex shape whose vertices are the midpoints of the sides of the tetrahedron. Show that $V\leq \frac{(U-|OA|-|BC| )(U-|OB|-|AC| )(U-|OC|-|AB| )}{(2^{7} \cdot 3)}$.
1990 Turkey Team Selection Test
The circles $k_1, k_2, k_3$ with radii ($a>c>b$) $a,b,c$ are tangent to line $d$ at $A,B,C$, respectively. $k_1$ is tangent to $k_2$, and $k_2$ is tangent to $k_3$. The tangent line to $k_3$ at $E$ is parallel to $d$, and it meets $k_1$ at $D$. The line perpendicular to $d$ at $A$ meets line $EB$ at $F$. Prove that $AD=AF$.
1990 Turkey Team Selection Test
For real numbers $x_i$, the statement $$ x_1 + x_2 + x_3 = 0 \Rightarrow x_1x_2 + x_2x_3 + x_3x_1 \leq 0$$ is always true. (Prove!)

For which $n\geq 4$ integers, the statement $$x_1 + x_2 + \dots + x_n = 0 \Rightarrow x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1 \leq 0$$ is always true. Justify your answer.
1990 Turkey Team Selection Test
Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$.

We define $$d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |$$ for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$.

Show that $n=23$ if $T$ has a subset $S$ satisfying



$|S|=2^k$

For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$
1990 Turkey Team Selection Test
Let $ABCD$ be a convex quadrilateral such that $$\begin{array}{rl} E,F \in [AB],& AE  =  EF = FB \\ G,H \in [BC],& BG  =  GH = HC \\ K,L \in [CD],& CK  =  KL = LD \\ M,N \in [DA],& DM  =  MN = NA  \end{array}$$ Let $$[NG] \cap [LE] = \{P\},  [NG]\cap [KF] = \{Q\},$$ $${[}MH] \cap [KF] = \{R\},  [MH]\cap [LE]=\{S\}$$

Prove that



$Area(ABCD) = 9 \cdot Area(PQRS)$

$NP=PQ=QG$
1990 Turkey Team Selection Test
Let $b_m$ be numbers of factors $2$ of the number $m!$ (that is, $2^{b_m}|m!$ and $2^{b_m+1}\nmid m!$). Find the least $m$ such that $m-b_m = 1990$.
1990 Turkey Team Selection Test
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.
1989 Turkey Team Selection Test
Let $\mathbb{Z}^+$ denote the set of positive integers. Find all functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that



$f(m,m)=m$

$f(m,k) = f(k,m)$

$f(m, m+k) = f(m,k)$



, for each $m,k \in \mathbb{Z}^+$.
1989 Turkey Team Selection Test
A positive integer is called a "double number" if its decimal representation consists of a block of digits, not commencing with $0$, followed immediately by an identical block. So, for instance, $360360$ is a double number, but $36036$ is not. Show that there are infinitely many double numbers which are perfect squares.
1989 Turkey Team Selection Test
Let $C_1$ and $C_2$ be given circles. Let $A_1$ on $C_1$ and $A_2$ on $C_2$ be fixed points. If chord $A_1P_1$ of $C_1$ is parallel to chord $A_2P_2$ of $C_2$, find the locus of the midpoint of $P_1P_2$.
1989 Turkey Team Selection Test
There is a stone on each square of $n\times n$ chessboard. We gather $n^2$ stones and distribute them to the squares (again each square contains one stone)  such that any two adjacent stones are again adjacent. Find all distributions such that at least one stone at the corners remains at its initial square. (Two squares are adjacent if they share a common edge.)
1989 Turkey Team Selection Test
There are $n\geq2$ weights such that each weighs a positive integer less than $n$ and their total weights is less than $2n$. Prove that there is a subset of these weights such that their total weights is equal to $n$.
1989 Turkey Team Selection Test
The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .
2018 Ukraine Team Selection Test
Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
2018 Ukraine Team Selection Test
Consider the set of all integer points in $Z^3$. Sasha and Masha play such a game. At first, Masha marks an arbitrary point. After that, Sasha marks all the points on some a plane perpendicular to one of the coordinate axes and at no point, which Masha noted. Next, they continue to take turns (Masha can't to select previously marked points, Sasha cannot choose the planes on which there are points said Masha). Masha wants to mark $n$ consecutive points on some line that parallel to one of the coordinate axes, and Sasha seeks to interfere with it. Find all $n$, in which Masha can achieve the desired result.
2018 Ukraine Team Selection Test
Let $n$ be an odd integer. Consider a square lattice of size $n \times  n$, consisting of $n^2$ unit squares and $2n(n +1)$ edges. All edges are painted in red or blue so that the number of red edges does not exceed $n^2$. Prove that there is a cell that has at least three blue edges.
2018 Ukraine Team Selection Test
Find the smallest positive number $\lambda$ such that for an arbitrary $12$ points on the plane $P_1,P_2,...P_{12}$ (points may coincide), with distance between arbitrary two of them does not exceeds $1$, holds the inequality $\sum_{1\le i\le j\le 12} P_iP_j^2 \le \lambda$
2018 Ukraine Team Selection Test
The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which  $n^2+d$ is the square of the natural number.

.
2018 Ukraine Team Selection Test
Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.
2018 Ukraine Team Selection Test
Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.
2018 Ukraine Team Selection Test
$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition honest, if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.
2018 Ukraine Team Selection Test
Let $n$ be a positive integer and $a_1,a_2,\dots,a_n$ be integers. Function $f: \mathbb{Z} \rightarrow \mathbb{R}$ is such that for all integers $k$ and $l$, $l \neq 0$, $$\sum_{i=1}^n f(k+a_il)=0.$$Prove that $f \equiv 0$.
2017 Ukraine Team Selection Test
Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration.



Proposed by Denys Smirnov
2017 Ukraine Team Selection Test
Whether exist set $A$ that contain 2016 real numbers (some of them may be equal) not all of which equal 0 such that next statement holds. For arbitrary 1008-element subset of $A$ there is a monic polynomial of degree 1008 such that elements of this subset are roots of the polynomial and other 1008 elements of $A$ are coefficients of this polynomial's degrees from 0 to 1007.
2017 Ukraine Team Selection Test
Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.
2017 Ukraine Team Selection Test
There're two positive inegers $a_1<a_2$. For every positive integer $n \geq 3$ let $a_n$ be the smallest integer that bigger than $a_{n-1}$ and such that there's unique pair $1\leq i< j\leq n-1$ such that this number equals to $a_i+a_j$. Given that there're finitely many even numbers in this sequence. Prove that sequence $\{a_{n+1}-a_n \}$ is periodic starting from some element.
2016 Ukraine Team Selection Test
Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called holes. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy?



Proposed by Bohdan Rublov
2016 Ukraine Team Selection Test
Find all functions from positive integers to itself such that $f(a+b)=f(a)+f(b)+f(c)+f(d)$ for all $c^2+d^2=2ab$
2016 Ukraine Team Selection Test
Let $ABC$ be an equilateral triangle of side $1$. There are three grasshoppers sitting in $A$, $B$, $C$. At any point of time for any two grasshoppers separated by a distance $d$ one of them can jump over other one so that distance between them becomes $2kd$, $k,d$ are nonfixed positive integers. Let $M$, $N$ be points on rays $AB$, $AC$ such that $AM=AN=l$, $l$ is fixed positive integer. In a finite number of jumps all of grasshoppers end up sitting inside the triangle $AMN$. Find, in terms of $l$, the number of final positions of the grasshoppers. (Grasshoppers can leave the triangle $AMN$ during their jumps.)
2016 Ukraine Team Selection Test
Let $a_1,\ldots, a_n$ be real numbers. Define polynomials $f,g$ by $$f(x)=\sum_{k=1}^n a_kx^k,\  g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.$$Assume that $g(2016)=0$. Prove that $f(x)$ has a root in $(0;2016)$.
2016 Ukraine Team Selection Test
Suppose that $a_0, a_1, \cdots $ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and $$ a_{n+1} = \gcd{(a_n, b_n)} + 1, \qquad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1. $$Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.
2015 Ukraine Team Selection Test
Let $O$ be the circumcenter of the  triangle $ABC, A'$ be a point symmetric of $A$ wrt line $BC, X$ is an arbitrary point on the ray $AA'$ ($X \ne A$). Angle bisector of angle $BAC$ intersects the circumcircle of triangle $ABC$ at point $D$ ($D \ne A$). Let $M$ be the midpoint of the segment $DX$. A line passing through point $O$ parallel to $AD$, intersects $DX$ at point $N$. Prove that angles $BAM$ and $CAN$ angles are equal.
2015 Ukraine Team Selection Test
In a football tournament, $n$ teams play one round ($n \vdots 2$). In each round should play $n / 2$ pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural $k$ such that the following situation is possible:

after $k$ tours, making a schedule of $k + 1$ rounds already is not possible,

i.e. these $n$ teams cannot be divided into $n / 2$ pairs, in each of which there are teams that have not played in the previous $k$ rounds.



PS. The 3 vertical dots notation in the first row, I do not know what it means.
2015 Ukraine Team Selection Test
A prime number $p> 3$ is given. Prove that integers less than $p$, it is possible to divide them into two non-empty sets such that the sum of the numbers in the first set will be congruent  modulo p to the product of the numbers in the second set.
2015 Ukraine Team Selection Test
Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines  $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.
2015 Ukraine Team Selection Test
Let $A$ and $B$ be two sets of real numbers. Suppose that the elements of the set $AB = \{ab: a\in A, b\in B\}$ form a finite arithmetic progression. Prove that one of these sets contains no more than three elements
2015 Ukraine Team Selection Test
Find all functions $f: R \to  R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$
2015 Ukraine Team Selection Test
The set $M$ consists of $n$ points on the plane and satisfies the conditions:

$\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon,

$\bullet$  for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon.

Find the smallest possible value that $n$ can take .
2015 Ukraine Team Selection Test
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.

(Here we always assume that an angle bisector is a ray.)



Proposed by Sergey Berlov, Russia
2015 Ukraine Team Selection Test
For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge  2015$ the set $A$ contains a three-element arithmetic sequence.
2014 Ukraine Team Selection Test
Given an integer $n \ge  2$ and a regular $2n$-polygon at each vertex of which sitting on an ant. At some points in time, each ant creeps into one of two adjacent peaks (some peaks may have several ants at a time). Through $k$ such operations, it turned out to be an arbitrary line connecting two different ones the vertices of a polygon with ants do not pass through its center. For given $n$ find the lowest possible  value of $k$.
2014 Ukraine Team Selection Test
Let $x_1,x_2,\cdots,x_n$ be postive real numbers such that $x_1x_2\cdots x_n=1$ ,$S=x^3_1+x^3_2+\cdots+x^3_n$.Find the maximum of $\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}$
2014 Ukraine Team Selection Test
Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.