Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Bosnia And Herzegovina - Regional Olympiad
$h_a$, $h_b$ and  $h_c$ are altitudes,  $t_a$, $t_b$ and  $t_c$ are medians of acute triangle, $r$ radius of incircle, and $R$ radius of circumcircle of acute triangle $ABC$. Prove that $$\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}$$
2016 Bosnia And Herzegovina - Regional Olympiad
It is given circle with center in center of coordinate center with radius of $2016$. On circle and inside it are $540$ points  with integer coordinates such that no three of them are collinear. Prove that there exist two triangles with vertices in given points such that they have same area
2016 Bosnia And Herzegovina - Regional Olympiad
Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that

$a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$

$b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$
2016 Bosnia And Herzegovina - Regional Olympiad
Find all elements $n \in A = \{2,3,...,2016\} \subset \mathbb{N}$ such that:

every number $m \in A$ smaller than $n$, and coprime with $n$, must be a prime number
2016 Bosnia And Herzegovina - Regional Olympiad
Circle of radius $R_1$ is inscribed in an acute angle $\alpha$. Second circle with radius $R_2$ touches one of the sides forming the angle $\alpha$ in same point as first circle and intersects the second side in points $A$ and $B$, such that centers of both circles lie inside angle $\alpha$. Prove that $$AB=4\cos{\frac{\alpha}{2}}\sqrt{(R_2-R_1)\left(R_1 \cos^2 \frac{\alpha}{2}+R_2 \sin^2 \frac{\alpha}{2}\right)}$$
2016 Bosnia And Herzegovina - Regional Olympiad
Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that:

$a)$ $f(1)+2>0$

$b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$

$c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$
2015 Bosnia And Herzegovina - Regional Olympiad
Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$
2015 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$
2015 Bosnia And Herzegovina - Regional Olympiad
In parallelogram $ABCD$ holds $AB=BD$. Let $K$ be a point on $AB$, different from $A$, such that $KD=AD$. Let $M$ be a point symmetric to $C$ with respect to $K$, and $N$ be a point symmetric to point $B$ with respect to $A$. Prove that $DM=DN$
2015 Bosnia And Herzegovina - Regional Olympiad
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
2015 Bosnia And Herzegovina - Regional Olympiad
Solve the inequation: $$5\mid x\mid \leq x(3x+2-2\sqrt{8-2x-x^2})$$
2015 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality:

$$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$
2015 Bosnia And Herzegovina - Regional Olympiad
Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$
2015 Bosnia And Herzegovina - Regional Olympiad
On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points.

$a)$ Prove that there exist two students with exactly the same answers to problems

$b)$ Prove that there exist at least $4$ students with same number of points
2015 Bosnia And Herzegovina - Regional Olympiad
For real numbers $x$, $y$ and $z$, solve the system of equations:

$$x^3+y^3=3y+3z+4$$$$y^3+z^3=3z+3x+4$$$$x^3+z^3=3x+3y+4$$
2015 Bosnia And Herzegovina - Regional Olympiad
Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$is prime
2015 Bosnia And Herzegovina - Regional Olympiad
Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$
2015 Bosnia And Herzegovina - Regional Olympiad
There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:



i.) Each pair of students are in exactly one club.



ii.) For each student and each society, the student is in exactly one club of the society.



iii.) Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is

in exactly $m$ societies.



Find all possible values of $k$.



Proposed by Guihua Gong, Puerto Rico
2015 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality:

$$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$
2015 Bosnia And Herzegovina - Regional Olympiad
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
2015 Bosnia And Herzegovina - Regional Olympiad
Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$
2015 Bosnia And Herzegovina - Regional Olympiad
It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that:

$a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled

$b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled

Which numbers must be expelled such that sum of numbers remaining in set stays minimal
2014 Bosnia And Herzegovina - Regional Olympiad
Find all possible values of $$\frac{(a+b-c)^2}{(a-c)(b-c)}+\frac{(b+c-a)^2}{(b-a)(c-a)}+\frac{(c+a-b)^2}{(c-b)(a-b)}$$
2014 Bosnia And Herzegovina - Regional Olympiad
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$
2014 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ $(b \geq c)$, point $E$ is the midpoint of shorter arc $BC$. If $D$ is the point such that $ED$ is the diameter of circumcircle $ABC$, prove that $\angle DEA = \frac{1}{2}(\beta-\gamma)$
2014 Bosnia And Herzegovina - Regional Olympiad
Determine the set $S$ with minimal number of points defining $7$ distinct lines
2014 Bosnia And Herzegovina - Regional Olympiad
Solve the equation: $$ \frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}=3$$where $x$, $y$ and $z$ are integers
2014 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$
2014 Bosnia And Herzegovina - Regional Olympiad
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.

Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.



Alternative formulation. Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.

Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
2014 Bosnia And Herzegovina - Regional Olympiad
How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.
2014 Bosnia And Herzegovina - Regional Olympiad
Solve logarithmical equation $x^{\log _{3} {x-1}} + 2(x-1)^{\log _{3} {x}}=3x^2$
2014 Bosnia And Herzegovina - Regional Olympiad
Solve the equation $$x^2+y^2+z^2=686$$where $x$, $y$ and $z$ are positive integers
2014 Bosnia And Herzegovina - Regional Olympiad
Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$
2014 Bosnia And Herzegovina - Regional Olympiad
At the beginning of school year in one of the first grade classes:

$i)$ every student had exatly $20$ acquaintances

$ii)$ every two students knowing each other had exactly $13$ mutual acquaintances

$iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances

Find number of students in this class
2014 Bosnia And Herzegovina - Regional Olympiad
Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$$$y=\frac{2x^2}{1+x^2}$$$$z=\frac{2y^2}{1+y^2}$$
2014 Bosnia And Herzegovina - Regional Olympiad
Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers
2013 Bosnia And Herzegovina - Regional Olympiad
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$
2013 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$, $\angle ACB=50^{\circ}$ and $\angle CBA=70^{\circ}$. Let $D$ be a foot of perpendicular from point $A$ to side $BC$, $O$ circumcenter of $ABC$ and $E$ antipode of $A$ in circumcircle $ABC$. Find $\angle DAE$
2013 Bosnia And Herzegovina - Regional Olympiad
Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers
2013 Bosnia And Herzegovina - Regional Olympiad
$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$

$b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$
2013 Bosnia And Herzegovina - Regional Olympiad
If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$
2013 Bosnia And Herzegovina - Regional Olympiad
In circle with radius $10$, point $M$ is on chord $PQ$ such that $PM=5$ and $MQ=10$. Through point $M$ we draw chords $AB$ and $CD$, and points $X$ and $Y$ are intersection points of chords $AD$ and $BC$ with chord $PQ$ (see picture), respectively. If $XM=3$ find $MY$
2013 Bosnia And Herzegovina - Regional Olympiad
Find all integers $a$ such that $\sqrt{\frac{9a+4}{a-6}}$ is rational number
2013 Bosnia And Herzegovina - Regional Olympiad
Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$if and only if $a=b$
2013 Bosnia And Herzegovina - Regional Olympiad
Find all integers $a$, $b$, $c$ and $d$ such that $$a^2+5b^2-2c^2-2cd-3d^2=0$$
2013 Bosnia And Herzegovina - Regional Olympiad
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
2013 Bosnia And Herzegovina - Regional Olympiad
If $A=\{1,2,...,4s-1,4s\}$ and $S \subseteq A$ such that $\mid S \mid =2s+2$, prove that in $S$ we can find three distinct numbers $x$, $y$ and $z$ such that $x+y=2z$
2013 Bosnia And Herzegovina - Regional Olympiad
If $a$, $b$ and $c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$, prove that $$\frac{1}{2} \leq \frac{a}{1+a^4}+\frac{b}{1+b^4}+\frac{c}{1+c^4} \leq \frac{9\sqrt{3}}{10}$$
2013 Bosnia And Herzegovina - Regional Olympiad
If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer
2012 Bosnia And Herzegovina - Regional Olympiad
Find all possible values of $$\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$$where $a$, $b$ and $c$ are positive real numbers such that $ab+bc+ca=abc$
2012 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$
2012 Bosnia And Herzegovina - Regional Olympiad
Find remainder when dividing upon $2012$ number $$A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011$$
2012 Bosnia And Herzegovina - Regional Olympiad
Let $S$ be an incenter of triangle $ABC$ and let incircle touch sides $AC$ and $AB$ in points $P$ and $Q$, respectively. Lines $BS$ and $CS$ intersect line $PQ$ in points $M$ and $N$, respectively. Prove that points $M$, $N$, $B$ and $C$ are concyclic
2012 Bosnia And Herzegovina - Regional Olympiad
Solve equation $$x^2-\sqrt{a-x}=a$$where $x$ is real number and $a$ is real parameter
2012 Bosnia And Herzegovina - Regional Olympiad
Harry Potter can do any of the three tricks arbitrary number of times:

$i)$ switch $1$ plum and $1$ pear with $2$ apples

$ii)$ switch $1$ pear and $1$ apple with $3$ plums

$iii)$ switch $1$ apple and $1$ plum with $4$ pears

In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks and now he has $2012$ apples, $2012$ pears and more than $2012$ plums. What is the minimal number of plums he can have?
2012 Bosnia And Herzegovina - Regional Olympiad
Quadrilateral $ABCD$ is cyclic. Line through point $D$ parallel with line $BC$ intersects $CA$ in point $P$, line $AB$ in point $Q$ and circumcircle of $ABCD$ in point $R$. Line through point $D$ parallel with line $AB$ intersects $AC$ in point $S$, line $BC$ in point $T$ and circumcircle of $ABCD$ in point $U$. If $PQ=QR$, prove that $ST=TU$
2012 Bosnia And Herzegovina - Regional Olympiad
Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer
2012 Bosnia And Herzegovina - Regional Olympiad
For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$
2012 Bosnia And Herzegovina - Regional Olympiad
On football toornament there were $4$ teams participating. Every team played exactly one match with every other team. For the win, winner gets $3$ points, while if draw both teams get $1$ point. If at the end of tournament every team had different number of points and first place team had $6$ points, find the points of other teams
2012 Bosnia And Herzegovina - Regional Olympiad
Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$
2012 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ point $O$ is circumcenter. Point $T$ is centroid of $ABC$, and points $D$, $E$ and $F$ are circumcenters of triangles $TBC$, $TCA$ and $TAB$. Prove that $O$ is centroid of $DEF$
2012 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$
2012 Bosnia And Herzegovina - Regional Olympiad
Prove the inequality: $$\frac{A+a+B+b}{A+a+B+b+c+r}+\frac{B+b+C+c}{B+b+C+c+a+r}>\frac{C+c+A+a}{C+c+A+a+b+r}$$where $A$, $B$, $C$, $a$, $b$, $c$ and $r$ are positive real numbers
2011 Bosnia And Herzegovina - Regional Olympiad
Factorise $$(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3$$
2011 Bosnia And Herzegovina - Regional Olympiad
At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$
2011 Bosnia And Herzegovina - Regional Olympiad
Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$
2011 Bosnia And Herzegovina - Regional Olympiad
For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$is not perfect square
2011 Bosnia And Herzegovina - Regional Olympiad
Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$
2011 Bosnia And Herzegovina - Regional Olympiad
If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$Prove that $p$ divides $m$
2011 Bosnia And Herzegovina - Regional Olympiad
Let $I$ be the incircle and $O$ a circumcenter  of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$
2011 Bosnia And Herzegovina - Regional Olympiad
Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$

$a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$

$b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$
2011 Bosnia And Herzegovina - Regional Olympiad
Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$has root in interval $\left(0,\frac{\pi}{2}\right)$
2011 Bosnia And Herzegovina - Regional Olympiad
For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$
2011 Bosnia And Herzegovina - Regional Olympiad
Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$
2011 Bosnia And Herzegovina - Regional Olympiad
Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational
2011 Bosnia And Herzegovina - Regional Olympiad
If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$
2011 Bosnia And Herzegovina - Regional Olympiad
If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$
2010 Bosnia And Herzegovina - Regional Olympiad
For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$$$a^3+b^3+c^3+d^3=0$$Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero
2010 Bosnia And Herzegovina - Regional Olympiad
In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic
2010 Bosnia And Herzegovina - Regional Olympiad
If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$
2010 Bosnia And Herzegovina - Regional Olympiad
In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times
2010 Bosnia And Herzegovina - Regional Olympiad
Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$$$y-\frac{x+3y}{x^2+y^2}=0$$
2010 Bosnia And Herzegovina - Regional Olympiad
It is given acute triangle $ABC$  with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$
2010 Bosnia And Herzegovina - Regional Olympiad
It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that

$i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements

$ii)$ Upper bound can be reached for $n=3$
2010 Bosnia And Herzegovina - Regional Olympiad
Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$where $x$, $y$ and $z$ are real numbers
2010 Bosnia And Herzegovina - Regional Olympiad
Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$
2010 Bosnia And Herzegovina - Regional Olympiad
Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$
2010 Bosnia And Herzegovina - Regional Olympiad
In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points
2010 Bosnia And Herzegovina - Regional Olympiad
It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$$$ 2<a^2<3$$Find the value of $$a^{12}-\frac{144}{a}$$
2010 Bosnia And Herzegovina - Regional Olympiad
Let $AA_1$, $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ and let $A_1A_2$, $B_1B_2$ and $C_1C_2$ be  diameters of Euler circle of triangle $ABC$. Prove that lines $AA_2$, $BB_2$ and $CC_2$ are concurrent
2009 Bosnia And Herzegovina - Regional Olympiad
Find all triplets of integers $(x,y,z)$ such that $$xy(x^2-y^2)+yz(y^2-z^2)+zx(z^2-x^2)=1$$
2009 Bosnia And Herzegovina - Regional Olympiad
Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$
2009 Bosnia And Herzegovina - Regional Olympiad
Is it possible in a plane mark $10$ red, $10$ blue and $10$ green points (all distinct) such that three conditions hold:

$i)$   For every red point $A$ there exists a blue point closer to point $A$ than any other green point

$ii)$  For every blue point $B$ there exists a green point closer to point $B$ than any other red point

$iii)$ For every green point $C$ there exists a red point closer to point $C$ than any other blue point
2009 Bosnia And Herzegovina - Regional Olympiad
Let $C$ be a circle with center $O$ and radius $R$. From point $A$ of circle $C$ we construct a tangent $t$ on circle $C$. We construct line $d$ through point $O$ whch intersects tangent $t$ in point $M$ and circle $C$ in points $B$ and $D$ ($B$ lies between points $O$ and $M$). If $AM=R\sqrt{3}$, prove:

$a)$ Triangle $AMD$ is isosceles

$b)$ Circumcenter of $AMD$ lies on circle $C$
2009 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ such that $\angle ACB=90^{\circ}$, let point $H$ be foot of perpendicular from point $C$ to side $AB$. Show that sum of radiuses of incircles of $ABC$, $BCH$ and $ACH$ is $CH$
2009 Bosnia And Herzegovina - Regional Olympiad
Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$$$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$has solution in set of real numbers
2009 Bosnia And Herzegovina - Regional Olympiad
Decomposition of number $n$ is showing $n$ as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer $n$ show that number of decompositions is $2^{n-1}$
2009 Bosnia And Herzegovina - Regional Olympiad
Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers  $\frac{x^2-1}{y+1}$ and  $\frac{y^2-1}{x+1}$ are integers