Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 Bosnia Herzegovina Team Selection Test
Let $N$ be a positive integer. It is given set of weights which satisfies following conditions:

$i)$ Every weight from set has some weight from $1,2,...,N$;

$ii)$ For every $i\in {1,2,...,N}$ in given set there exists weight $i$;

$iii)$ Sum of all weights from given set is even positive integer.

Prove that set can be partitioned into two disjoint sets which have equal weight
2015 Bosnia Herzegovina Team Selection Test
Let $D$, $E$ and $F$ be points in which incircle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$, respectively, and let $I$ be a center of that circle.Furthermore, let $P$ be a foot of perpendicular from point $I$ to line $AD$, and let $M$ be midpoint of $DE$. If $\{N\}=PM\cap{AC}$, prove that $DN \parallel EF$
2014 Bosnia Herzegovina Team Selection Test
Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$
2014 Bosnia Herzegovina Team Selection Test
Let $a$ ,$b$ and $c$ be distinct real numbers.

$a)$ Determine value of $ \frac{1+ab }{a-b} \cdot \frac{1+bc }{b-c} + \frac{1+bc }{b-c} \cdot \frac{1+ca }{c-a} + \frac{1+ca }{c-a} \cdot \frac{1+ab}{a-b}  $



$b)$ Determine value of $ \frac{1-ab }{a-b} \cdot \frac{1-bc }{b-c} + \frac{1-bc }{b-c} \cdot \frac{1-ca }{c-a} + \frac{1-ca }{c-a} \cdot \frac{1-ab}{a-b}  $



$c)$ Prove the following ineqaulity

$ \frac{1+a^2b^2 }{(a-b)^2} + \frac{1+b^2c^2 }{(b-c)^2} + \frac{1+c^2a^2 }{(c-a)^2} \geq \frac{3}{2} $



When does eqaulity holds?
2014 Bosnia Herzegovina Team Selection Test
Find all nonnegative integer numbers such that $7^x- 2 \cdot 5^y = -1$
2014 Bosnia Herzegovina Team Selection Test
Sequence $a_n$ is defined by $a_1=\frac{1}{2}$, $a_m=\frac{a_{m-1}}{2m \cdot a_{m-1} + 1}$ for $m>1$. Determine value of $a_1+a_2+...+a_k$ in terms of $k$, where $k$ is positive integer.
2014 Bosnia Herzegovina Team Selection Test
It is given regular $n$-sided polygon, $n \geq 6$. How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
2014 Bosnia Herzegovina Team Selection Test
Let $D$ and $E$ be foots of altitudes from $A$ and $B$ of triangle $ABC$, $F$ be intersection point of angle bisector from $C$ with side $AB$, and $O$, $I$ and $H$ be circumcenter, center of inscribed circle and orthocenter of triangle $ABC$, respectively. If $\frac{CF}{AD}+ \frac{CF}{BE}=2$, prove that $OI = IH$.
2013 Bosnia Herzegovina Team Selection Test
Triangle $ABC$ is right angled at $C$.  Lines $AM$ and $BN$ are internal angle bisectors.

$AM$ and $BN$ intersect altitude $CH$ at points $P$ and $Q$ respectively.

Prove that the line which passes through the midpoints of segments $QN$ and $PM$  is parallel to $AB$.
2013 Bosnia Herzegovina Team Selection Test
The sequence $a_n$ is defined by $a_0=a_1=1$ and $a_{n+1}=14a_n-a_{n-1}-4$,for all positive integers $n$.

Prove that all terms of this sequence are perfect squares.
2013 Bosnia Herzegovina Team Selection Test
Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.
2013 Bosnia Herzegovina Team Selection Test
Find all primes $p,q$ such that $p$ divides $30q-1$ and $q$ divides $30p-1$.
2013 Bosnia Herzegovina Team Selection Test
Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$.

Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$.

Find:

a) $\min F_3$;

b) $\min F_4$;

c) $\min F_5$.
2013 Bosnia Herzegovina Team Selection Test
In triangle $ABC$, $I$ is the incenter.  We have chosen points $P,Q,R$ on segments $IA,IB,IC$ respectively such that $IP\cdot IA=IQ \cdot IB=IR\cdot IC$.

Prove that the points $I$ and $O$ belong to Euler line of triangle $PQR$ where $O$ is circumcenter of $ABC$.
2012 Bosnia Herzegovina Team Selection Test
Let $D$ be the midpoint of the arc $B-A-C$ of the circumcircle of $\triangle ABC (AB<AC)$. Let $E$ be the foot of perpendicular from $D$ to $AC$. Prove that $|CE|=\frac{|BA|+|AC|}{2}$.
2012 Bosnia Herzegovina Team Selection Test
Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$:



$$\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.$$
2012 Bosnia Herzegovina Team Selection Test
Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that:



$$mp=x_1^2+x_2^2+x_3^2.$$
2012 Bosnia Herzegovina Team Selection Test
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, $$f(1)=p+1,$$ $$f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,$$ where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.
2012 Bosnia Herzegovina Team Selection Test
Given is a triangle $\triangle ABC$ and points $M$ and $K$ on lines $AB$ and $CB$ such that $AM=AC=CK$. Prove that the length of the radius of the circumcircle of triangle $\triangle BKM$ is equal to the lenght $OI$, where $O$ and $I$ are centers of the circumcircle and the incircle of $\triangle ABC$, respectively. Also prove that $OI\perp MK$.
2012 Bosnia Herzegovina Team Selection Test
A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.
2011 Bosnia Herzegovina Team Selection Test
In triangle $ABC$ it holds $|BC|= \frac{1}{2}(|AB|+|AC|)$. Let $M$ and $N$ be midpoints of $AB$ and $AC$, and let $I$ be the incenter of $ABC$. Prove that $A, M, I, N$ are concyclic.
2011 Bosnia Herzegovina Team Selection Test
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$
2011 Bosnia Herzegovina Team Selection Test
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number

$$W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|$$

is a perfect square.
2011 Bosnia Herzegovina Team Selection Test
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.
2011 Bosnia Herzegovina Team Selection Test
Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality

$$a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1$$

holds.
2011 Bosnia Herzegovina Team Selection Test
In quadrilateral $ABCD$ sides $AD$ and $BC$ aren't parallel. Diagonals $AC$ and $BD$ intersect in $E$. $F$ and $G$ are points on sides $AB$ and $DC$ such $\frac{AF}{FB}=\frac{DG}{GC}=\frac{AD}{BC}$ Prove that if $E, F, G$ are collinear then $ABCD$ is cyclic.
2010 Bosnia Herzegovina Team Selection Test
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$.

$b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.
2010 Bosnia Herzegovina Team Selection Test
Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.
2010 Bosnia Herzegovina Team Selection Test
Find all functions $ f :\mathbb{Z}\mapsto\mathbb{Z} $ such that following conditions holds:

$a)$ $f(n) \cdot f(-n)=f(n^2)$ for all $n\in\mathbb{Z}$

$b)$ $f(m+n)=f(m)+f(n)+2mn$ for all $m,n\in\mathbb{Z}$
2010 Bosnia Herzegovina Team Selection Test
Let $a$,$b$ and $c$ be sides of a triangle such that $a+b+c\le2$. Prove that

$-3<{\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-\frac{a^3}{c}-\frac{b^3}{a}-\frac{c^3}{b}}<3$
2010 Bosnia Herzegovina Team Selection Test
Prove that total number of ones which is showed in all nonrestricted partitions of natural number $n$ is equal to sum of numbers of distinct elements in that partitions.
2009 Bosnia Herzegovina Team Selection Test
Denote by $M$ and $N$ feets of perpendiculars from $A$ to angle bisectors of exterior angles at $B$ and $C,$ in triangle $\triangle ABC.$ Prove that the length of segment $MN$ is equal to semiperimeter of triangle $\triangle ABC.$
2009 Bosnia Herzegovina Team Selection Test
Find all pairs $\left(a,b\right)$ of posive integers such that $\frac{a^{2}\left(b-a\right)}{b+a}$ is square of prime.
2009 Bosnia Herzegovina Team Selection Test
$a_{1},a_{2},\dots,a_{100}$ are real numbers such that:$$

a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0$$

$$

a_{1}^{2}+a_{2}^{2}\geq100$$

$$

a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100$$

What is the minimum value of sum $a_{1}+a_{2}+\dots+a_{100}.$
2009 Bosnia Herzegovina Team Selection Test
Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?
2009 Bosnia Herzegovina Team Selection Test
Line $p$ intersects sides $AB$ and $BC$ of triangle $\triangle ABC$ at points $M$ and $K.$ If area of triangle $\triangle MBK$ is equal to area of quadrilateral $AMKC,$ prove that $$\frac{\left|MB\right|+\left|BK\right|}{\left|AM\right|+\left|CA\right|+\left|KC\right|}\geq\frac{1}{3}$$
2009 Bosnia Herzegovina Team Selection Test
Let $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that $$

\left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor $$
2008 Bosnia Herzegovina Team Selection Test
Prove that in an isosceles  triangle  $ \triangle ABC$ with $ AC=BC=b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.
2008 Bosnia Herzegovina Team Selection Test
Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions:



(i) $ m^{2}-n$ divides $ m+n^{2}$



(ii) $ n^{2}-m$ divides $ n+m^{2}$
2008 Bosnia Herzegovina Team Selection Test
$ 30$ persons are sitting at round table. $ 30 - N$ of them always speak true ("true speakers") while the other $ N$ of them sometimes speak true sometimes not ("lie speakers"). Question: "Who is your right neighbour - "true speaker" or "lie speaker" ?" is asked to all 30 persons and 30 answers are collected. What is maximal number $ N$ for which (with knowledge of these answers) we can always be sure (decide) about at least one person who is "true speaker".
2008 Bosnia Herzegovina Team Selection Test
$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.
2008 Bosnia Herzegovina Team Selection Test
Let $ AD$ be height of triangle $ \triangle ABC$ and $ R$ circumradius. Denote by $ E$ and $ F$ feet of perpendiculars from point $ D$ to sides $ AB$ and $ AC$.



If $ AD=R\sqrt{2}$, prove that circumcenter of triangle $ \triangle ABC$ lies on line $ EF$.
2008 Bosnia Herzegovina Team Selection Test
Find all functions $ f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying

$$ f(f(x) + y) = f(x^2 - y) + 4f(x)y

$$

for all $ x,y \in \mathbb{R}$.
2007 Bosnia Herzegovina Team Selection Test
Let $ABC$ be a triangle such that length of internal angle bisector from $B$ is equal to $s$. Also, length of external angle bisector from $B$ is equal to $s_1$. Find area of triangle $ABC$ if $\frac{AB}{BC} = k$
2007 Bosnia Herzegovina Team Selection Test
Find all pairs of integers $(x,y)$ such that

$x(x+2)=y^2(y^2+1)$
2007 Bosnia Herzegovina Team Selection Test
Find all $ x\in \mathbb{Z} $ and $ a\in \mathbb{R} $ satisfying

$$\sqrt{x^2-4}+\sqrt{x+2} = \sqrt{x-a}+a $$
2007 Bosnia Herzegovina Team Selection Test
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?
2007 Bosnia Herzegovina Team Selection Test
Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN$. Let $T$ be the intersection point of angle bisector of $\angle ABC$ and line from $K$ perpendicular to $AN$. Determine value of  $\angle MTB$.
2007 Bosnia Herzegovina Team Selection Test
The set $A$ has exactly $n>4$ elements. Ann chooses $n+1$ distinct subsets of $A$, such that every subset has exactly $3$ elements. Prove that there exist two subsets chosen by Ann which have exactly one common element.
2006 Bosnia and Herzegovina Team Selection Test
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?
2006 Bosnia and Herzegovina Team Selection Test
It is given a triangle $\triangle ABC$. Determine the locus of center of rectangle inscribed in triangle $ABC$ such that one side of rectangle lies on side $AB$.
2006 Bosnia and Herzegovina Team Selection Test
Prove that for every positive integer $n$ holds inequality $\{n\sqrt{7}\}>\frac{3\sqrt{7}}{14n}$, where $\{x\}$ is fractional part of $x$.
2006 Bosnia and Herzegovina Team Selection Test
Prove that every infinite arithmetic progression $a$, $a+d$, $a+2d$,... where $a$ and $d$ are positive integers, contains infinte geometric progression $b$, $bq$, $bq^2$,... where $b$ and $q$ are also positive integers
2006 Bosnia and Herzegovina Team Selection Test
Triangle $ABC$ is inscribed in circle with center $O$. Let $P$ be a point on arc $AB$ which does not contain point $C$. Perpendicular from point $P$ on line $BO$ intersects side $AB$ in point $S$, and side $BC$ in $T$. Perpendicular from point $P$ on line $AO$ intersects side $AB$ in point $Q$, and side $AC$ in $R$.

(i) Prove that triangle $PQS$ is isosceles

(ii) Prove that $\frac{PQ}{QR}=\frac{ST}{PQ}$
2006 Bosnia and Herzegovina Team Selection Test
Let $a_1$, $a_2$,...,$a_n$ be constant real numbers and $x$ be variable real number $x$. Let $f(x)=cos(a_1+x)+\frac{cos(a_2+x)}{2}+\frac{cos(a_3+x)}{2^2}+...+\frac{cos(a_n+x)}{2^{n-1}}$. If $f(x_1)=f(x_2)=0$, prove that $x_1-x_2=m\pi$, where $m$ is integer.
2005 Bosnia and Herzegovina Team Selection Test
Let $H$ be an orthocenter of an acute triangle $ABC$. Prove that midpoints of $AB$ and $CH$ and intersection point of angle bisectors of $\angle CAH$ and $\angle CBH$ lie on the same line.
2005 Bosnia and Herzegovina Team Selection Test
If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$
2005 Bosnia and Herzegovina Team Selection Test
Let $n$ be a positive integer such that $n \geq 2$. Let $x_1, x_2,..., x_n$ be $n$ distinct positive integers and $S_i$ sum of all numbers between them except $x_i$ for $i=1,2,...,n$. Let $f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.$

Determine maximal value of $f(x_1,x_2,...,x_n)$, while $(x_1,x_2,...,x_n)$ is an element of set which consists from all $n$-tuples of distinct positive integers.
2005 Bosnia and Herzegovina Team Selection Test
On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.
2005 Bosnia and Herzegovina Team Selection Test
If for an arbitrary permutation $(a_1,a_2,...,a_n)$ of set ${1,2,...,n}$ holds $\frac{{a_k}^2}{a_{k+1}}\leq k+2$,

$k=1,2,...,n-1$, prove that $a_k=k$ for $k=1,2,...,n$
2005 Bosnia and Herzegovina Team Selection Test
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
2004 Bosnia and Herzegovina Team Selection Test
Circle $k$ with center $O$ is touched from inside by two circles in points $S$ and $T,$ respectively. Let those two circles intersect at points $M$ and $N$, such that $N$ is closer to line $ST$. Prove that $OM$ and $MN$ are perpendicular iff $S$, $N$ and $T$ are collinear
2004 Bosnia and Herzegovina Team Selection Test
Determine whether does exists a triangle with area $2004$ with his sides positive integers.
2004 Bosnia and Herzegovina Team Selection Test
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality:

$\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$
2004 Bosnia and Herzegovina Team Selection Test
On competition which has $16$ teams, it is played $55$ games. Prove that among them exists $3$ teams such that they have not played any matches between themselves.
2004 Bosnia and Herzegovina Team Selection Test
For $0 \leq x < \frac{\pi}{2} $ prove the inequality:

$a^2\tan(x)\cdot(\cos(x))^{\frac{1}{3}}+b^2\sin{x}\geq 2xab$

where $a$ and $b$ are real numbers.
2004 Bosnia and Herzegovina Team Selection Test
It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent
2003 Bosnia and Herzegovina Team Selection Test
Board has written numbers: $5$, $7$ and $9$. In every step we do the following: for every pair $(a,b)$, $a>b$ numbers from the board, we also write the number $5a-4b$. Is it possible that after some iterations, $2003$ occurs at the board ?
2003 Bosnia and Herzegovina Team Selection Test
Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.
2003 Bosnia and Herzegovina Team Selection Test
Prove that for every positive integer $n$ holds:

$(n-1)^n+2n^n \leq (n+1)^{n} \leq 2(n-1)^n+2n^{n}$
2003 Bosnia and Herzegovina Team Selection Test
In triangle $ABC$ $AD$ and $BE$ are altitudes. Let $L$ be a point on $ED$ such that $ED$ is orthogonal to $BL$. If $LB^2=LD\cdot LE$ prove that triangle $ABC$ is isosceles
2003 Bosnia and Herzegovina Team Selection Test
It is given regular polygon with $2n$ sides and center $S$. Consider every quadrilateral with vertices as vertices of polygon. Let $u$ be number of such quadrilaterals which contain point $S$ inside and $v$ number of remaining quadrilaterals. Find $u-v$
2003 Bosnia and Herzegovina Team Selection Test
Let $a$, $b$ and $c$ be real numbers such that $\mid a \mid >2$ and $a^2+b^2+c^2=abc+4$. Prove that numbers $x$ and $y$ exist such that $a=x+\frac{1}{x}$, $b=y+\frac{1}{y}$ and $c=xy+\frac{1}{xy}$.
2002 Bosnia Herzegovina Team Selection Test
Let $x,y,z$ be real numbers that satisfy

$$x+y+z= 3 \ \ \text{ and } \ \ xy+yz+zx= a$$where $a$ is a real parameter. Find the value of $a$ for which the difference between the maximum and minimum possible values of $x$ equals $8$.
2002 Bosnia Herzegovina Team Selection Test
Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.
2002 Bosnia Herzegovina Team Selection Test
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
2002 Bosnia Herzegovina Team Selection Test
Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that

$$\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab} \ge \frac35$$
2002 Bosnia Herzegovina Team Selection Test
The vertices of the convex quadrilateral $ABCD$ and the intersection point $S$ of its diagonals are integer points in the plane. Let $P$ be the area of $ABCD$ and $P_1$ the area of triangle $ABS$. Prove that

$$\sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2$$
2002 Bosnia Herzegovina Team Selection Test
Let $p$ and $q$ be different prime numbers. Solve the following system in integers:

$$\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.$$
2001 Bosnia and Herzegovina Team Selection Test
On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$
2001 Bosnia and Herzegovina Team Selection Test
For positive integers $x$, $y$ and $z$ holds $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$.

Prove that $xyz\geq 3600$
2001 Bosnia and Herzegovina Team Selection Test
Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$
2001 Bosnia and Herzegovina Team Selection Test
In plane there  are two circles with radiuses $r_1$ and $r_2$, one outside the other. There are two external common tangents on those circles and one internal common tangent. The internal one intersects external ones in points $A$ and $B$ and touches one of the circles in point $C$. Prove that

$AC \cdot BC=r_1\cdot r_2$
2001 Bosnia and Herzegovina Team Selection Test
Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$
2001 Bosnia and Herzegovina Team Selection Test
Prove that there exists infinitely many positive integers $n$ such that equation $(x+y+z)^3=n^2xyz$ has solution $(x,y,z)$ in set $\mathbb{N}^3$
2000 Bosnia and Herzegovina Team Selection Test
Find real roots $x_1$, $x_2$ of equation $x^5-55x+21=0$, if we know $x_1\cdot x_2=1$
2000 Bosnia and Herzegovina Team Selection Test
Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$iff $S$ is incenter of $ABC$
2000 Bosnia and Herzegovina Team Selection Test
We call Pythagorean triple a triple $(x,y,z)$ of positive integers such that $x<y<z$ and $x^2+y^2=z^2$. Prove that for all $n \in \mathbb{N}$ number $2^{n+1}$ is in exactly $n$ Pythagorean triples
2000 Bosnia and Herzegovina Team Selection Test
Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$
2000 Bosnia and Herzegovina Team Selection Test
Let $T_m$ be a number of non-congruent triangles which perimeter is $m$ and all its sides are positive integers. Prove that:

$a)$ $T_{1999} > T_{2000}$

$b)$ $T_{4n+1}=T_{4n-2}+n$, $(n \in \mathbb{N})$