Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Azerbaijan Team Selection Test
Find all polynomials $P(x)$ with real coefficents, such that for all $x,y,z$ satisfying $x+y+z=0$, the equation below is true: $$P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))$$
2016 Azerbaijan Team Selection Test
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by $$ a_0 = M + \frac{1}{2}   \qquad  \textrm{and} \qquad    a_{k+1} = a_k\lfloor a_k \rfloor   \quad \textrm{for} \, k = 0, 1, 2, \cdots $$contains at least one integer term.
2016 Azerbaijan Team Selection Test
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2016 Azerbaijan Team Selection Test
Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that $$f(f(x)+y)=f(x)+3x+yf(y)$$for all positive reals $x,y$.
2016 Azerbaijan Team Selection Test
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies $$a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}$$for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
2016 Azerbaijan Team Selection Test
A positive interger $n$ is called rising if its decimal representation $a_ka_{k-1}\cdots a_0$ satisfies the condition $a_k\le a_{k-1}\le\cdots \le a_0$. Polynomial $P$ with real coefficents is called interger-valued if for all integer numbers $n$, $P(n)$ takes interger values. $P(n)$ is called rising-valued if for all rising numbers $n$, $P(n)$ takes integer values.

Does it necessarily mean that, "every rising-valued $P$ is also interger-valued $P$"?
2016 Azerbaijan Team Selection Test
During a day $2016$ customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer $k$ that makes the following statement always true.



We can find $k$ customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.
2016 Azerbaijan TST First Round.
Find the maximum value of natural components of number $96$ that we can seperate such that all of them must be relatively prime number withh each other.
2016 Azerbaijan TST First Round.
$ABC$ be atriangle with sides $AB=20$ , $AC=21$ and $BC=29$. Let $D$ and $E$ be points on the side $BC$ such that $BD=8$ and $EC=9$. Find the angle $\angle DAE$.
2016 Azerbaijan TST First Round.
Find the solution of the equation $8x(2x^2-1)(8x^4-8x^2+1)=1$ in the interval $[0,1]$?
2016 Azerbaijan TST First Round.
Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$



P.S: $P(y)=y^{2015}$ is also a function with power $2015$
2016 Azerbaijan TST First Round.
The largest side of the triangle $ABC$ is equal to $1$ unit. Prove that , the circles centred at $A,B$ and $C$ wit radiuses $\frac{1}{\sqrt{3}}$ can compeletely cover the triangle $ABC$.
2015 Azerbaijan Team Selection Test
Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$. The lines tangent to $\omega$ at the points $A$ and $B$ meet at $K$. The line passing through $K$ and parallel to $BC$ intersects the side $AC$ at $S$. Prove that $BS=CS$
2015 Azerbaijan Team Selection Test
Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.
2015 Azerbaijan IMO TST
We say that $A$$=${$a_1,a_2,a_3\cdots a_n$} consisting $n>2$ distinct positive integers is $good$ if for every $i=1,2,3\cdots n$ the number ${a_i}^{2015}$ is divisible by the product of all numbers in $A$ except $a_i$. Find all integers $n>2$ such that exists a $good$ set consisting of $n$ positive integers.
2015 Azerbaijan IMO TST
Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality $$(x-y)^2\leq|f(x) -f(y)|\leq|x-y|$$ is satisfied for all $x,y\in [0,1]$
2015 Azerbaijan IMO TST
Consider a trapezoid $ABCD$ with $BC||AD$ and $BC<AD$. Let the lines $AB$ and $CD$ meet at $X$. Let $\omega_1$ be the incircle of the triangle $XBC$, and let  $\omega_2$ be the excircle of the triangle $XAD$ which is tangent to the segment $AD$ . Denote by $a$ and $d$ the lines tangent to $\omega_1$ , distinct from $AB$ and $CD$, and passing through $A$ and $D$, respectively. Denote by $b$ and $c$ the lines tangent to $\omega_2$ , distinct from $AB$ and $CD$, passing through $B$ and $C$ respectively. Assume that the lines $a,b,c$ and $d$ are distinct. Prove that they form a parallelogram.
2015 Azerbaijan IMO TST
Let $n$ and $k$ be two positive integers such that $n>k$. Prove that the equation $x^n+y^n=z^k$ has a solution in positive integers if and only if the equation $x^n+y^n=z^{n-k}$ has a solution in positive integers.
2019 Belarus Team Selection Test
Does there exist a function $f:\mathbb N\to\mathbb N$ such that

$$

f(f(n+1))=f(f(n))+2^{n-1}

$$for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.)



(I. Gorodnin)
2019 Belarus Team Selection Test
Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that

$$

AB+CD>\max(AM+DM,BN+CN)?

$$

(Folklore)
2019 Belarus Team Selection Test
Given the equation

$$

a^b\cdot b^c=c^a

$$in positive integers $a$, $b$, and $c$.

(i) Prove that any prime divisor of $a$ divides $b$ as well.

(ii) Solve the equation under the assumption $b\ge a$.

(iii) Prove that the equation has infinitely many solutions.



(I. Voronovich)
2019 Belarus Team Selection Test
Let the sequence $(a_n)$ be constructed in the following way:

$$

a_1=1,\mbox{ }a_2=1,\mbox{ }a_{n+2}=a_{n+1}+\frac{1}{a_n},\mbox{ }n=1,2,\ldots.

$$Prove that $a_{180}>19$.



(Folklore)
2019 Belarus Team Selection Test
Given a quadratic trinomial $p(x)$ with integer coefficients such that $p(x)$ is not divisible by $3$ for all integers $x$.

Prove that there exist polynomials $f(x)$ and $h(x)$ with integer coefficients such that

$$

p(x)\cdot f(x)+3h(x)=x^6+x^4+x^2+1.

$$

(I. Gorodnin)
2019 Belarus Team Selection Test
Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$.

Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$.



(M. Berindeanu, RMC 2018 book)
2019 Belarus Team Selection Test
$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box.

Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones.



(O. Izhboldin)
2019 Belarus Team Selection Test
Cells of $11\times 11$ table are colored with $n$ colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than $7$ and not greater than $13$.

Prove that there exists at least one row or column which contains cells of at least four different colors.



(N. Sedrakyan)
2019 Belarus Team Selection Test
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.
2019 Belarus Team Selection Test
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.



Proposed by Mongolia
2019 Belarus Team Selection Test
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called good, if the following conditions hold:



each triangle from $T$ is inscribed in $\omega$;

no two triangles from $T$ have a common interior point.



Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
2019 Belarus Team Selection Test
Four positive integers $x,y,z$ and $t$ satisfy the relations

$$ xy - zt = x + y = z + t. $$Is it possible that both $xy$ and $zt$ are perfect squares?
2019 Belarus Team Selection Test
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1\geq k \geq n$ satisfying $$a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}.$$Find the maximum possible value of $a_{2018}-a_{2017}$.
2019 Belarus Team Selection Test
A function $f:\mathbb N\to\mathbb N$, where $\mathbb N$ is the set of positive integers, satisfies the following condition: for any positive integers $m$ and $n$ ($m>n$) the number $f(m)-f(n)$ is divisible by $m-n$.

Is the function $f$ necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial $p(x)$ with real coefficients such that $f(n)=p(n)$ for all positive integers $n$?)



(Folklore)
2019 Belarus Team Selection Test
Let $AA_1$ be the bisector of a triangle $ABC$. Points $D$ and $F$ are chosen on the line $BC$ such that $A_1$ is the midpoint of the segment $DF$. A line $l$, different from $BC$, passes through $A_1$ and intersects the lines $AB$ and $AC$ at points $B_1$ and $C_1$, respectively.

Find the locus of the points of intersection of the lines $B_1D$ and $C_1F$ for all possible positions of $l$.



(M. Karpuk)
2019 Belarus Team Selection Test
A polygon (not necessarily convex) on the coordinate plane is called plump if it satisfies the following conditions:

$\bullet$ coordinates of vertices are integers;

$\bullet$ each side forms an angle of $0^\circ$, $90^\circ$, or $45^\circ$ with the abscissa axis;

$\bullet$ internal angles belong to the interval $[90^\circ, 270^\circ]$.

Prove that if a square of each side length of a plump polygon is even, then such a polygon can be cut into several convex plump polygons.



(A. Yuran)
2019 Belarus Team Selection Test
Two circles $\Omega$ and $\Gamma$ are internally tangent at the point $B$. The chord $AC$ of $\Gamma$ is tangent to $\Omega$ at the point $L$, and the segments $AB$ and $BC$ intersect $\Omega$ at the points $M$ and $N$. Let $M_1$ and $N_1$ be the reflections of $M$ and $N$ about the line $BL$; and let $M_2$ and $N_2$ be the reflections of $M$ and $N$ about the line $AC$. The lines $M_1M_2$ and $N_1N_2$ intersect at the point $K$.

Prove that the lines $BK$ and $AC$ are perpendicular.



(M. Karpuk)
2019 Belarus Team Selection Test
The numbers $1,2,\ldots,49,50$ are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers $a,b,c$ from the board, replaces them by their sum $a+b+c$ and writes $(a+b)(b+c)(c+a)$ to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all $24$ numbers written in the notebook. Let $A$ and $B$ be the maximum and the minimum possible sums that Ann san obtain.

Find the value of $\frac{A}{B}$.



(I. Voronovich)
2019 Belarus Team Selection Test
Let $n \ge 2018$ be an integer, and let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be pairwise distinct positive integers not exceeding $5n$. Suppose that the sequence

$$ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n} $$forms an arithmetic progression. Prove that the terms of the sequence are equal.
2019 Belarus Team Selection Test
The internal bisectors of angles $\angle DAB$ and $\angle BCD$ of a quadrilateral $ABCD$ intersect at the point $X_1$, and the external bisectors of these angles intersect at the point $X_2$. The internal bisectors of angles $\angle ABC$ and $\angle CDA$ intersect at the point $Y_1$, and the external bisectors of these angles intersect at the point $Y_2$.

Prove that the angle between the lines $X_1X_2$ and $Y_1Y_2$ equals the angle between the diagonals $AC$ and $BD$.



(A. Voidelevich)
2019 Belarus Team Selection Test
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
2019 Belarus Team Selection Test
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that

$$

(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.

$$

(B. Serankou, M. Karpuk)
2019 Belarus Team Selection Test
Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions:

1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$;

2. $f$ takes all integer values.



(I. Voronovich)
2019 Belarus Team Selection Test
Prove that for $n>1$ , $n$ does not divide $2^{n-1}+1$
2016 Belarus Team Selection Test
Prove for positive $a,b,c$ that

$$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$
2016 Belarus Team Selection Test
Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively

Find the angles of the triangle $ABC$
2016 Belarus Team Selection Test
Solve the equation $p^3-q^3=pq^3-1$ in primes $p,q$.
2016 Belarus Team Selection Test
There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.

What is the smallest number of the edges in the initial graph with 30 vertices?
2016 Belarus Team Selection Test
Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively.

Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$
2016 Belarus Team Selection Test
Find all real numbers $a$ such that exists function $\mathbb {R} \rightarrow \mathbb {R} $ satisfying the following conditions:

1) $f(f(x)) =xf(x)-ax$ for all real $x$

2) $f$ is not constant

3) $f$ takes the value $a$
2016 Belarus Team Selection Test
Point $A,B$ are marked on the right branch of the hyperbola  $y=\frac{1}{x},x>0$. The straight line $l$ passing through the origin $O$ is perpendicular to $AB$ and meets  $AB$ and given branch of the hyperbola at points $D$ and $C$ respectively. The circle through $A,B,C$ meets $l$ at $F$.

Find $OD:CF$
2016 Belarus Team Selection Test
Find all functions $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ such that

$$f(x-2f(y))= xf(y)-yf(x)+g(x)$$for all real $x,y$
2016 Belarus Team Selection Test
Given a graph with $n \geq 4$ vertices. It is known that for any two of vertices there is a vertex connected with none of these two vertices.

Find the greatest possible number of the edges in the graph.
2016 Belarus Team Selection Test
Let $D,E,F$  denote the tangent points of the incircle of $ABC$ with sides $BC,AC,AB$ respectively. Let $M$ be the midpoint  of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D,M,F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D,M,E$ and the segment $AC$.

Prove that the circle passing through $A,K,L$ touches the line $BC$
2016 Belarus Team Selection Test
Given real numbers $a,b,c,d$ such that $\sin{a}+b >\sin{c}+d, a+\sin{b}>c+\sin{d}$, prove that $a+b>c+d$
2016 Belarus Team Selection Test
A point $A_1$ is marked inside an acute non-isosceles triangle $ABC$ such that $\angle A_1AB = \angle A_1BC$ and $\angle A_1AC=\angle A_1CB$. Points $B_1$ and $C_1$ are defined same way. Let $G$ be the gravity center if the triangle $ABC$.

Prove that the points $A_1,B_1,C_1,G$ are concyclic.
2016 Belarus Team Selection Test
Solve the equation $2^a-5^b=3$ in positive integers $a,b$.
2016 Belarus Team Selection Test
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
2016 Belarus Team Selection Test
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is clean if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
2016 Belarus Team Selection Test
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that$$f(x-f(y))=f(f(x))-f(y)-1$$holds for all $x,y\in\mathbb{Z}$. (It is 2015 IMO Shortlist A2 )

b) The same question for if $$f(x-f(y))=f(f(x))-f(y)-2$$for all integers $x,y$
2016 Belarus Team Selection Test
Points $B_1$ and $C_1$ are marked respectively on the sides $AB$ and $AC$ of an acute isosceles triangle $ABC$( $AB=AC$) such that $BB_1=AC_1$.  The points $B,C$ and $S$ lie in the same half-plane with respect to the line $B_1C_1$ so that $\angle SB_1C_1=\angle SC_1B_1 = \angle BAC$

Prove that $B,C,S$ are colinear if and only if the triangle $ABC$ is equilateral.
2016 Belarus Team Selection Test
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
2016 Belarus Team Selection Test
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2016 Belarus Team Selection Test
There are $n\geq1$ cities on a horizontal line. Each city is guarded by a pair of stationary elephants, one just to the left and one just ot the right of the city, and facing away from it. The $2n$ elephants are of different sizes. If an elephant walks forward, it will knock aside any elephant that it approaches from behind, and  in face-to-face meeting, the smaller elephant will be knocked aside. A moving elephant will keep walking in the same direction until it is knocked aside.

Show that there is a unique city with the property that if any of the other cities orders its elephants to walk, then that city will not be invaded by an elephant.



IMO 2015, Shortlist C1, modified by G. Smith
2016 Belarus Team Selection Test
Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $B_1$ and $C_1$ be the midpoints of the sides $AC$ and $AB$ respectively Let $M$ and $N$ be symmetric to $B$ and $C$ about $B_1$ and $C_1$ respectively.

Prove that the lines $KM$ and $LN$ meet on $BC$.
2016 Belarus Team Selection Test
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2015 Belarus Team Selection Test
Solve the equation in nonnegative integers $a,b,c$:



$3^a+2^b+2015=3c!$



I.Gorodnin
2015 Belarus Team Selection Test
All the numbers $1,2,...,9$ are written in the cells of a $3\times 3$ table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary $2\times2$ square of the table and either decrease by $1$ or increase by $1$ all four numbers of the square. After some number of such moves all numbers of the table become equal to some number $a$. Find all possible values of $a$.



I.Voronovich
2015 Belarus Team Selection Test
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $P$ and $Q$ respectively. $N$ and $M$ are the midpoints of $AC$ and $BC$ respectively. Let $X=AM\cap BP, Y=BN\cap AQ$. Given $C,X,Y$ are collinear, prove that $CX$ is the angle bisector of the angle $ACB$.



I. Gorodnin
2015 Belarus Team Selection Test
Prove that $(a+b+c)^5 \ge 81 (a^2+b^2+c^2)abc$ for any positive real numbers $a,b,c$



I.Gorodnin
2015 Belarus Team Selection Test
N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$.



I.Gorodnin
2015 Belarus Team Selection Test
In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?



Folklore
2015 Belarus Team Selection Test
Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quasrilateral.



I.Gorodnin
2015 Belarus Team Selection Test
Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$.



I.Voronovich
2015 Belarus Team Selection Test
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$?



I.Voronovich
2015 Belarus Team Selection Test
The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$.

If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$.



I. Gorodnin
2015 Belarus Team Selection Test
Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.



Proposed by Serbia
2015 Belarus Team Selection Test
A  circle intersects a parabola at four distinct points. Let $M$ and $N$ be the midpoints of the arcs of the circle which are outside the parabola. Prove that the line $MN$ is perpendicular to the axis of the parabola.



I. Voronovich
2015 Belarus Team Selection Test
Determine all pairs $(x, y)$ of positive integers such that $$\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.$$

Proposed by Titu Andreescu, USA
2015 Belarus Team Selection Test
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.

Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.



Proposed by Tamas Fleiner and Peter Pal Pach, Hungary
2015 Belarus Team Selection Test
Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$.



B.Gilevich