Navvye/Olympiad-Problems-CSV huggingface dataset:
2000 Belarus Team Selection Test
Starting with an arbitrary pair (a,b) of vectors on the plane, we are allowed to perform the operations of the following two types:

(1) To replace $(a,b)$ with $(a+2kb,b)$ for an arbitrary integer $k \ne 0$;

(2) To replace $(a,b)$ with $(a,b+2ka)$ for an arbitrary integer $ k \ne 0$.

However, we must change the type of operetion in any step.

(a) Is it possible to obtain $((1,0), (2,1))$ from $((1,0), (0,1))$, if the first operation is of the type (1)?

(b) Find all pairs of vectors that can be obtained from $((1,0), (0,1))$ (the type of first operation can be selected arbitrarily).
2000 Belarus Team Selection Test
For any positive numbers $a,b,c,x,y, z$, prove the inequality $ \frac{a^3}{x}+ \frac{b^3}{y}+ \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$
2000 Belarus Team Selection Test
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
2000 Belarus Team Selection Test
A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice nice if at the end nobody has her own ball and it is called tiresome if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.
2000 Belarus Team Selection Test
The diagonals of a convex quadrilateral $ABCD$ with $AB = AC = BD$ intersect at $P$, and $O$ and $I$ are the circumcenter and incenter of $\vartriangle ABP$, respectively. Prove that if $O \ne I$ then $OI$ and $CD$ are perpendicular
2000 Belarus Team Selection Test
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
2000 Belarus Team Selection Test
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.
1998 Belarus Team Selection Test
Do there exist functions $f : R  \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$for all real $x$ ?
1998 Belarus Team Selection Test
Find all finite sets $M \subset R$ containing at least two elements such that $(2a/3 -b^2) \in M$ for any two different elements $a,b \in M$.
1998 Belarus Team Selection Test
Let $s,t$ be given nonzero integers, $(x,y)$ be any (ordered) pair of integers. A sequence of moves is performed as follows: per move $(x,y)$ changes to $(x+t, y-s)$. The pair (x,y) is said to be good  if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime.

a) Determine whether $(s,t)$ is itself a good pair;

bj Prove that for any nonzero $s$ and $t$ there is a pair $(x,y)$ which is not good.
1998 Belarus Team Selection Test
The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$
1998 Belarus Team Selection Test
Any of $6$ gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?
1998 Belarus Team Selection Test
a) Given that integers $a$ and  $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.

b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
1998 Belarus Team Selection Test
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle.

Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
1998 Belarus Team Selection Test
Prove the inequality $$\sum_{k=1}^{n}\frac{\sin (k+1)x}{\sin kx}< 2\frac{\cos x}{\sin^2x}$$where $0 < nx < \pi/2$, $n \in  N$.
1998 Belarus Team Selection Test
Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$).

Prove that $(S(n))^3 <n^4$.
1998 Belarus Team Selection Test
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n - 1$ boys $ b_1, b_2, \ldots, b_{2n-1}.$ The girl $ g_i,$ $ i = 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i-1},$ and no others. For all $ r = 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) = B(r)$ for each $ r = 1, 2, \ldots, n.$
1998 Belarus Team Selection Test
Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i = 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i+1}$ and $ A_iA_{i+2}$ (the addition of indices being mod 3). Let $ B_i, i = 1, 2, 3,$ be the second point of intersection of $ C_{i+1}$ and $ C_{i+2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
1998 Belarus Team Selection Test
For each finite set $ U$ of nonzero vectors in the plane we define $ l(U)$ to be the length of the vector that is the sum of all vectors in $ U.$ Given a finite set $ V$ of nonzero vectors in the plane, a subset $ B$ of $ V$ is said to be maximal if $ l(B)$ is greater than or equal to $ l(A)$ for each nonempty subset $ A$ of $ V.$



(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.



(b) Show that, for any set $ V$ consisting of $ n \geq 1$ vectors the number of maximal subsets is less than or equal to $ 2n.$
1998 Belarus Team Selection Test
For any sequence of real numbers $(a_n), n \in N$, define a new sequence $(b_n)$ as $b_n =a_{n+2}+sa_{n+1}+ta_{n}$, where $s,t$ are given real numbers.

Find all ordered pairs $(s,t)$ satisfying the following property: any sequence $(a_n)$ converges as soon as the sequence $(b_n)$ converges.
1998 Belarus Team Selection Test
a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4  + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer.



b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?
1998 Belarus Team Selection Test
The lengths of the sides of a convex hexagon $ ABCDEF$ satisfy $ AB = BC$, $ CD = DE$, $ EF = FA$. Prove that:

$$ \frac {BC}{BE} + \frac {DE}{DA} + \frac {FA}{FC} \geq \frac {3}{2}.

$$
1998 Belarus Team Selection Test
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
1998 Belarus Team Selection Test
Let $1=d_1<d_2<d_3<...<d_k=n$ be all different divisors of positive integer $n$  written in ascending order. Determine all $n$  such that $$d_7^2+d_{10}^2=(n/d_{22})^2.$$
1998 Belarus Team Selection Test
Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$.



Ya. Konstantinovski
1998 Belarus Team Selection Test
Let $a$, $b$, $c$ be real positive numbers. Show that $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1$$
1998 Belarus Team Selection Test
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules:  $ R_1 = (1),$ and if $ R_{n - 1} = (x_1, \ldots, x_s),$ then



$$ R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).$$



For example, $ R_2 = (1, 2),$ $ R_3 = (1, 1, 2, 3),$ $ R_4 = (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
1998 Belarus Team Selection Test
Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.
1998 Belarus Team Selection Test
The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$.



I. Voronovich
1998 Belarus Team Selection Test
Find all continuous functions $f: R \to R$  such that $g(g(x)) = g(x)+2x$ for all real $x$.
1998 Belarus Team Selection Test
Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$.



S.Shikh
1998 Belarus Team Selection Test
The numbers $1,2,...,n$ ($n \ge 5$) are written on the circle in the clockwise order. Per move it is allowed to exchange any couple of consecutive numbers $a, b$ to the couple  $\frac{a+b}{2}, \frac{a+b}{2}$.

Is it possible to make all numbers equal using these operations?
1998 Belarus Team Selection Test
For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows:

a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle  A_1B_1C_1$  consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on.

Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.
1995 Belarus Team Selection Test
There is a 100 x100 square table, a real number being written in each cell.$A$ and $B$ play the following game. They choose, turn by turn, some row of the table (if it has not been chosen before). When $A$ and $B$ have $50$ rows chosen each, they sum the numbers in the corresponding cells of the chosen rows, and then sum the squares of all $100$ obtained numbers and compare the results. $A$ player who has the greater result wins. Player $A$ begins. Show that $A$ can avoid a defeat.
1995 Belarus Team Selection Test
Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$
1995 Belarus Team Selection Test
Show that there is no infinite sequence an of natural numbers such that $$a_{a_n}=a_{n+1}a_{n-1}-a_{n}^2$$for all $n\geq 2$
1995 Belarus Team Selection Test
Prove that the number of odd coefficients in the polynomial $(1+x)^n$ is a power of $2$ for every positive integer $N$
1995 Belarus Team Selection Test
There is a room having a form of right-angled parallelepiped. Four maps of the same scale are hung (generally, on different levels over the floor) on four walls of the room, so that sides of the maps are parallel to sides of the wall. It is known that the four points corresponding to each of Stockholm, Moscow, and Istanbul are coplanar. Prove that the four points coresponding to Hong Kong are coplanar as well.
1995 Belarus Team Selection Test
If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: $$a^pb^q+(1-a)^p(1-b)^q\le 1$$
2018 Bosnia and Herzegovina EGMO TST
$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$

$a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$
2018 Bosnia and Herzegovina EGMO TST
Prove that for every pair of positive integers $(m,n)$, bigger than $2$, there exists positive integer $k$ and numbers $a_0,a_1,...,a_k$, which are bigger than $2$, such that $a_0=m$, $a_1=n$ and for all $i=0,1,...,k-1$ holds

$$ a_i+a_{i+1} \mid a_ia_{i+1}+1$$
2018 Bosnia and Herzegovina EGMO TST
Let $O$ be a circumcenter of acute triangle $ABC$ and let $O_1$ and $O_2$ be circumcenters of triangles $OAB$ and $OAC$, respectively. Circumcircles of triangles $OAB$ and $OAC$ intersect side $BC$ in points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Perpendicular bisector of side $BC$ intersects side $AC$ in point $F$($F \neq A$). Prove that circumcenter of triangle $ADE$ lies on $AC$ iff $F$ lies on line $O_1O_2$
2018 Bosnia and Herzegovina EGMO TST
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$,  maximum number of separators
2017 Bosnia and Herzegovina EGMO TST
It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.
2017 Bosnia and Herzegovina EGMO TST
It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$
2017 Bosnia and Herzegovina EGMO TST
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$
2017 Bosnia and Herzegovina EGMO TST
Let $a$, $b$, $c$, $d$ and $e$ be distinct positive real numbers such that $a^2+b^2+c^2+d^2+e^2=ab+ac+ad+ae+bc+bd+be+cd+ce+de$

$a)$ Prove that among these $5$ numbers there exists triplet such that they cannot be sides of a triangle

$b)$ Prove that, for $a)$,  there exists at least $6$ different triplets
2018 Bosnia and Herzegovina Team Selection Test
In acute triangle $ABC$ $(AB < AC)$ let $D$, $E$ and $F$ be foots of perpedicular from $A$, $B$ and $C$ to $BC$, $CA$ and $AB$, respectively. Let $P$ and $Q$ be points on line $EF$ such that $DP \perp EF$ and $BQ=CQ$. Prove that $\angle ADP = \angle PBQ$
2018 Bosnia and Herzegovina Team Selection Test
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that

$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$If $M>1$, prove that the polynomial

$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$has no positive roots.
2018 Bosnia and Herzegovina Team Selection Test
Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.
2018 Bosnia and Herzegovina Team Selection Test
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$
2018 Bosnia and Herzegovina Team Selection Test
Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:

$$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$.

The goal of Eduardo is to make  $M$ divisible by $p$, and the goal of Fernando is to prevent this.



Prove that Eduardo has a winning strategy.



Proposed by Amine Natik, Morocco
2018 Bosnia and Herzegovina Team Selection Test
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2017 Bosnia Herzegovina Team Selection Test
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.



Proposed by Dorlir Ahmeti, Albania
2017 Bosnia Herzegovina Team Selection Test
Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.
2017 Bosnia Herzegovina Team Selection Test
There are $ 6n + 4$ mathematicians participating in a conference which includes $ 2n + 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time.

1. Determine whether or not there is the case $ n = 1$.

2. Determine whether or not there is the case $ n > 1$.
2017 Bosnia Herzegovina Team Selection Test
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that

$$ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. $$
2017 Bosnia Herzegovina Team Selection Test
Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.
2016 Bosnia and Herzegovina Team Selection Test
Let $ABCD$ be a quadrilateral inscribed in circle $k$. Lines $AB$ and $CD$ intersect at point $E$ such that $AB=BE$. Let $F$ be the intersection point of tangents on circle $k$ in points $B$ and $D$, respectively. If the lines $AB$ and $DF$ are parallel, prove that $A$, $C$ and $F$ are collinear.
2016 Bosnia and Herzegovina Team Selection Test
Let $n$ be a positive integer and let $t$ be an integer. $n$ distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to $t$. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the $4$ sentences:

$i.$ Is there a number on the table equal to $k$?

$ii.$ If a number $k$ exists on the table, erase him.

$iii.$ If a number $k$ does not exist on the table, add him.

$iv.$ Do the numbers written on the table can be arranged in two sets with equal sum of elements?

On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than $3n$ moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to $t$
2016 Bosnia and Herzegovina Team Selection Test
For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is nice if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements:



$a)$  If there is given a nice sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$.



$b)$ For every prime number $p>2$, there exist a nice sequence such that no terms of the sequence are divisible by $p$.
2016 Bosnia and Herzegovina Team Selection Test
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.
2016 Bosnia and Herzegovina Team Selection Test
Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear.

.
2016 Bosnia and Herzegovina Team Selection Test
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that $$f(x-f(y))=f(f(x))-f(y)-1$$holds for all $x,y\in\mathbb{Z}$.
2015 Bosnia Herzegovina Team Selection Test
Determine the minimum value of the expression

$$\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}$$

for positive real numbers $a,b,c$ such that $a+b+c \leq 3$.
2015 Bosnia Herzegovina Team Selection Test
Let $D$ be an arbitrary point on side $AB$ of triangle $ABC$. Circumcircles of triangles $BCD$ and $ACD$ intersect sides $AC$ and $BC$ at points $E$ and $F$, respectively. Perpendicular bisector of $EF$ cuts $AB$ at point $M$, and line perpendicular to $AB$ at $D$ at point $N$. Lines $AB$ and $EF$ intersect at point $T$, and the second point of intersection of circumcircle of triangle $CMD$ and line $TC$ is $U$. Prove that $NC=NU$
2015 Bosnia Herzegovina Team Selection Test
Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.
2015 Bosnia Herzegovina Team Selection Test
Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.