Navvye/Olympiad-Problems-CSV huggingface dataset:
1985 Austrian-Polish Competition
We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?
1985 Austrian-Polish Competition
Let $P$ be a point inside a tetrahedron $ABCD$ and let $S_A,S_B,S_C,S_D$ be the centroids (i.e. centers of gravity) of the tetrahedra $PBCD,PCDA,PDAB,PABC$. Show that the volume of the tetrahedron $S_AS_BS_CS_D$ equals $1/64$ the volume of $ABCD$.
1985 Austrian-Polish Competition
Find an upper bound for the ratio

$$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$

over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$.



Note. The smaller the bound, the better the solution.
1985 Austrian-Polish Competition
A convex $n$-gon $A_0A_1\dots A_{n-1}$ has been partitioned into $n-2$ triangles by certain diagonals not intersecting inside the $n$-gon. Prove that these triangles can be labeled $\triangle_1,\triangle_2,\dots,\triangle_{n-2}$ in such a way that $A_i$ is a vertex of $\triangle_i$, for $i=1,2,\dots,n-2$. Find the number of all such labellings.
1985 Austrian-Polish Competition
We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.
1984 Austrian-Polish Competition
Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.
1984 Austrian-Polish Competition
Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.
1984 Austrian-Polish Competition
Show that for $n>1$ and any positive real numbers $k,x_{1},x_{2},...,x_{n}$ then

$$\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}$$

Where $f(x)=k^x$. When does equality hold.
1984 Austrian-Polish Competition
A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$.

Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.
1984 Austrian-Polish Competition
Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$, find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and

$a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1$

$a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2$

$...$

$a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n$
1984 Austrian-Polish Competition
In a dancing hall, there are $n$ girls standing in one row and $n$ boys in the other row across them (so that all $2n$ dancers form a $2 \times  n$ board). Each dancer gives her / his left hand to a neighboring person standing to the left, across, or diagonally to the left. The analogous rule applies for right hands. No dancer gives both hands to the same person. In how many ways can the dancers do this?
1984 Austrian-Polish Competition
A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all$ j$. Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$.
1984 Austrian-Polish Competition
The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $ f_0(x) = 2x$ and$ f_1(x) =\frac{x}{x-1}$. Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k  \in  \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$.
1984 Austrian-Polish Competition
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$
1983 Austrian-Polish Competition
Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.
1983 Austrian-Polish Competition
Find all triples of positive integers $(p, q, n)$ with $p$ and $q$ prime, such that $p(p+1)+q(q+1) = n(n+1)$.
1983 Austrian-Polish Competition
A bounded planar region of area $S$ is covered by a finite family $F$ of closed discs. Prove that $F$ contains a subfamily consisting of pairwise disjoint discs, of joint area not less than $S/9$.
1983 Austrian-Polish Competition
The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in  N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.
1983 Austrian-Polish Competition
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system

$x_1 +  x_2 +  x_3  +  x_4 = 1 $

$a_1x_1 + a_2 x_2 +  a_3x_3  + a_4 x_4 = c$

$a_1^2x_1 + a_2^2 x_2 +  a_3^2x_3  + a_4^2 x_4 = c^2 $

has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
1983 Austrian-Polish Competition
Six straight lines are given in space. Among any three of them, two are perpendicular. Show that the given lines can be labeled $\ell_1,...,\ell_6$ in such a way that $\ell_1, \ell_2, \ell_3$ are pairwise perpendicular, and so are $\ell_4, \ell_5, \ell_6$.
1983 Austrian-Polish Competition
Let $P_1,P_2,P_3,P_4$ be four distinct points in the plane. Suppose $\ell_1,\ell_2, … , \ell_6$ are closed segments in that plane with the following property: Every straight line passing through at least one of the points $P_i$ meets the union $\ell_1 \cup \ell_2\cup …  \cup\ell_6$ in exactly two points. Prove or disprove that the segments $\ell_i$ necessarily form a hexagon.
1983 Austrian-Polish Competition
(a) Prove that $(2^{n+1}-1)!$   is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n

(b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show  that each $c_n$ is an integer.
1983 Austrian-Polish Competition
To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times  k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?
1982 Austrian-Polish Competition
Find all pairs $(n, m)$ of positive integers such that $gcd ((n + 1)^m - n, (n + 1)^{m+3} - n) > 1$.
1982 Austrian-Polish Competition
Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.
1982 Austrian-Polish Competition
If $n \ge 2$ is an integer, prove the equality

$$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$
1982 Austrian-Polish Competition
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
1982 Austrian-Polish Competition
Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.
1982 Austrian-Polish Competition
An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.
1982 Austrian-Polish Competition
Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties:

(i) $x^y = a^b = c^d$ and $x > a > c$

(ii) $z = ab = cd$

(iii) $x + y = a + b$.
1982 Austrian-Polish Competition
Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$, with equality only when $P$ is the centroid of $ABCD$.

Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.
1982 Austrian-Polish Competition
Define $S_n=\sum_{j,k=1}^{n} \frac{1}{\sqrt{j^2+k^2}}$.

Find a positive constant $C$ such that the inequality $n\le S_n \le Cn$  holds for all $n \ge 3$.

(Note. The smaller $C$, the better the solution.)
1981 Austrian-Polish Competition
Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.
1981 Austrian-Polish Competition
The sequence $a_0, a_1, a_2, ...$ is defined by $a_{n+1} = a^2_n + (a_n - 1)^2$ for $n \ge 0$. Find all rational numbers $a_0$ for which there exist four distinct indices $k, m, p, q$ such that $a_q - a_p = a_m - a_k$.
1981 Austrian-Polish Competition
Given is a triangle $ABC$, the inscribed circle $G$ of which has radius $r$. Let $r_a$ be the radius of the circle touching $AB$, $AC$ and $G$. [This circle lies inside triangle $ABC$.] Define $r_b$ and $r_c$ similarly. Prove that $r_a + r_b + r_c \geq r$ and find all cases in which equality occurs.



Bosnia - Herzegovina Mathematical Olympiad 2002
1981 Austrian-Polish Competition
Let $n \ge  3$ cells be arranged into a circle. Each cell can be occupied by $0$ or $1$. The following operation is admissible: Choose any cell $C$ occupied by a $1$, change it into a $0$ and simultaneously reverse the entries in the two cells adjacent to $C$ (so that $x,y$ become $1 - x$, $1 - y$). Initially, there is a $1$ in one cell and zeros elsewhere. For which values of $n$ is it possible to obtain zeros in all cells in a finite number of admissible steps?
1981 Austrian-Polish Competition
Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.
1981 Austrian-Polish Competition
The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$,$ y_{n+1}=z_n +\frac{1}{y_n}$,$z_{n+1}=x_n +\frac{1}{z_n} $ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.
1981 Austrian-Polish Competition
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
1981 Austrian-Polish Competition
The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?
1981 Austrian-Polish Competition
For a function $f : [0,1] \to [0,1] $ we define $f^1 = f $ and $f^{n+1} (x) = f (f^n(x))$ for $0 \le x \le 1$ and $n \in N$. Given that there is a  $n$ such that $|f^n(x) - f^n(y)| < |x - y| $ for all distinct $x, y \in [0,1]$, prove that there is a unique $x_0 \in [0,1]$ such that $f (x_0) = x_0$.
1980 Austrian-Polish Competition
Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.
1980 Austrian-Polish Competition
A sequence of integers $1 = x_1 < x_2 < x_3 <...$ satisfies $x_{n+1} \le  2n$ for all $n$. Show that every positive integer $k$ can be written as $x_j -x_i$ for some $i, j$.
1980 Austrian-Polish Competition
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.
1980 Austrian-Polish Competition
Prove that $\sum \frac{1}{i_1i_2 \ldots i_k} = n$ is taken over all non-empty subsets $\left\{i_1,i_2, \ldots, i_k\right\}$ of $\left\{1,2,\ldots,n\right\}$. (The $k$ is not fixed, so we are summing over all the $2^n-1$ possible nonempty subsets.)
1980 Austrian-Polish Competition
Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.
1980 Austrian-Polish Competition
Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality $$ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. $$Show that the following inequality holds for all positive integers $k,m$ $$ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. $$
1980 Austrian-Polish Competition
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: $$x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.$$
1980 Austrian-Polish Competition
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
1980 Austrian-Polish Competition
Through the endpoints $A$ and $B$ of a diameter $AB$ of a given circle, the tangents $\ell$  and $m$ have been drawn. Let $C\ne A$ be a point on $\ell$  and let $q_1,q_2$ be two rays from $C$. Ray $q_i$ cuts the circle in $D_i$ and $E_i$ with $D_i$ between $C$ and $E_i, i = 1,2$. Rays $AD_1,AD_2,AE_1,AE_2$ meet $m$ in the respective points $M_1,M_2,N_1,N_2$. Prove that $M_1M_2 = N_1N_2$.
1979 Austrian-Polish Competition
On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$. Let $BN$ be the altitude in triangle $BCE$. Prove that $\angle DNF = 90$.
1979 Austrian-Polish Competition
Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)$$with integer coefficients, having $n$ real roots $x_1,\dots,x_n$ satisfying $k \leq x_k \leq k+1$ for $k=1, \dots,n$.
1979 Austrian-Polish Competition
Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$holds for any $n$ positive numbers $a_1, \dots, a_n$.
1979 Austrian-Polish Competition
Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.
1979 Austrian-Polish Competition
The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.
1979 Austrian-Polish Competition
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $\sum_{i=1}^{n} x_i^k= a^k$ for $k = 1,2, ... ,n$.
1979 Austrian-Polish Competition
Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$, $B = (n,0)$, $C = (n,m)$, $D = (n-1,m)$, $E = (n-1,1)$, $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.
1979 Austrian-Polish Competition
Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$, respectively. Show that $AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$ .
1979 Austrian-Polish Competition
Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$, where $n$ is a given positive integer.
1978 Austrian-Polish Competition
Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy

$$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$
1978 Austrian-Polish Competition
A parallelogram is inscribed into a regular hexagon so that the centers of symmetry of both figures coincide. Prove that the area of the parallelogram does not exceed $2/3$ the area of the hexagon.
1978 Austrian-Polish Competition
Prove that

$$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$
1978 Austrian-Polish Competition
Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$, the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$.
1978 Austrian-Polish Competition
We are given $1978$ sets of size $40$ each. The size of the intersection of any two sets is exactly $1$. Prove that all the sets have a common element.
1978 Austrian-Polish Competition
We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.
1978 Austrian-Polish Competition
Let $M$ be the set of all lattice points in the plane (i.e. points with integer coordinates, in a fixed Cartesian coordinate system). For any point $P=(x,y)\in M$ we call the points $(x-1,y)$, $(x+1,y)$, $(x,y-1)$, $(x,y+1)$ neighbors of $P$. Let $S$ be a finite subset of $M$. A one-to-one mapping $f$ of $S$ onto $S$ is called perfect if $f(P)$ is a neighbor of $P$, for any $P\in S$. Prove that if such a mapping exists, then there exists also a perfect mapping $g:S\to S$ with the additional property $g(g(P))=P$ for $P\in S$.
1978 Austrian-Polish Competition
For any positive integer $k$ consider the sequence

$$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$

where there are $n$ square-root signs on the right-hand side.



(a) Show that the sequence converges, for every fixed integer $k\ge 1$.

(b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.
1978 Austrian-Polish Competition
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.
2016 Azerbaijan BMO TST
A line is called $good$ if it bisects perimeter and area of a figure at the same time.Prove that:



a) all of the good lines in a triangle concur.

b) all of the good lines in a regular polygon concur too.
2016 Azerbaijan BMO TST
Set $A$ consists of natural numbers such that these numbers can be expressed as $2x^2+3y^2,$ where $x$ and $y$ are integers. $(x^2+y^2\not=0)$

$a)$ Prove that there is no perfect square in the set $A.$

$b)$ Prove that multiple of odd number of elements of the set $A$ cannot be a perfect square.
2016 Azerbaijan BMO TST
$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$  presents have been accepted.
2016 Azerbaijan BMO TST
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(f(n))=n+2015$$where $n\in \mathbb{N}.$
2016 Azerbaijan BMO TST
Find all $n$ natural numbers such that for each of them there exist $p , q$ primes such that these terms satisfy.



$1.$ $p+2=q$

$2.$ $2^n+p$ and $2^n+q$ are primes.
2016 Azerbaijan BMO TST
İn triangle $ABC$ the bisector of $\angle BAC$ intersects the side $BC$ at the point $D$.The circle $\omega $ passes through $A$ and tangent to the side $BC$ at $D$.$AC$ and $\omega $ intersects at $M$ second time , $BM$ and $\omega $ intersects at $P$ second time. Prove that point $P$ lies on median of triangle $ABD$.
2016 Azerbaijan BMO TST
There are some checkers in $n\cdot n$ size chess board.Known that  for all numbers $1\le i,j\le n$  if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.
2016 Azerbaijan BMO TST
For all numbers $n\ge 1$ does there exist infinite positive numbers sequence $x_1,x_2,...,x_n$ such that $x_{n+2}=\sqrt{x_{n+1}}-\sqrt{x_n}$
2016 Azerbaijan BMO TST
Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.
2016 Azerbaijan BMO TST
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury

$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.

$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.
2016 Azerbaijan BMO TST
$a,b$ are positive integers and $(a!+b!)|a!b!$.Prove that $3a\ge 2b+2$.
2016 Azerbaijan BMO TST
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2017 Azerbaijan Team Selection Test
Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints:



each cell contains a distinct divisor;

the sums of all rows are equal; and

the sums of all columns are equal.
2017 Azerbaijan Team Selection Test
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
2017 Azerbaijan Team Selection Test
Find all functions $f : \mathbb R\to\mathbb R $ such that $$f(x+yf(x^2))=f(x)+xf(xy)$$for all real numbers $x$ and $y$.
2017 Azerbaijan Team Selection Test
Let $ABC$ be an acute angled triangle. Points $E$ and $F$ are chosen on the sides $AC$ and $AB$, respectively, such that $$BC^2=BA\times BF+CE\times CA.$$Prove that for all such $E$ and $F$, circumcircle of the triangle $AEF$ passes through a fixed point different from $A$.
2017 Azerbaijan Team Selection Test
Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers.

(a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$.

(b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.
2017 Azerbaijan Team Selection Test
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$, and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$. Show that the nu,erator  of the lowest term expression of each sum $x_1+x_2+...+x_k$ is a perfect square.
2017 Azerbaijan Team Selection Test
Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that



$m = 1$ and $l = 2k$; or

$l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$.
2017 Azerbaijan Team Selection Test
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2016 Azerbaijan Team Selection Test
Tangents from  the point $A$ to the circle $\Gamma$ touche this circle at $C$ and $D$.Let $B$ be a point on $\Gamma$,different from $C$ and $D$. The circle $\omega$ that passes through points $A$ and $B$ intersect with lines $AC$ and $AD$ at $F$ and $E$,respectively.Prove that the circumcircles of triangles $ABC$ and $DEB$ are tangent if and only if the points $C,D,F$ and $E$ are cyclic.