Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 Belarusian National Olympiad
For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number.
2018 Belarusian National Olympiad
Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the  grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.
2018 Belarusian National Olympiad
Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$has a solution in integers $a$ and $b$.
2018 Belarusian National Olympiad
The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$.
2018 Belarusian National Olympiad
The square $A_1B_1C_1D_1$ is inscribed in the right triangle $ABC$ (with $C=90$) so that points $A_1$, $B_1$ lie on the legs $CB$ and $CA$ respectively ,and points $C_1$, $D_1$ lie on the hypotenuse $AB$. The circumcircle of triangles $B_1A_1C$ an $AC_1B_1$ intersect at $B_1$ and $Y$. Prove that the lines $A_1X$ and $B_1Y$ meet on the hypotenuse $AB$.
2018 Belarusian National Olympiad
The vertices of the regular $n$-gon and its center are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center. The winner I a player if after his maveit is possible to get any marked point from any other moving along the segments. For each $n>2$ determine who has a winning strategy.
2018 Belarusian National Olympiad
Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality

$$

f(f(x))=x^2f(x)+ax^2

$$for all real $x$.
2018 Belarusian National Olympiad
The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$.

Prove that $\angle BXC=90^{\circ}$.
2018 Belarusian National Olympiad
For all pairs $(m, n)$ of positive integers that have the same number $k$ of divisors we define the operation $\circ$. Write all their divisors in an ascending order: $1=m_1<\ldots<m_k=m$, $1=n_1<\ldots<n_k=n$ and set

$$

m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k. 

$$Find all pairs of numbers $(m, n)$, $m\geqslant n$, such that $m\circ n=497$.
2018 Belarusian National Olympiad
A checkered polygon $A$ is drawn on the checkered plane. We call a cell of $A$ internal if all $8$ of its adjacent cells belong to $A$. All other (non-internal) cells of $A$ we call boundary. It is known that $1)$ each boundary cell has exactly two common sides with no boundary cells; and 2) the union of all boundary cells can be divided into isosceles trapezoid of area $2$ with vertices at the grid nodes (and acute angles of the trapezoids are equal $45^\circ$).

Prove that the area of the polygon $A$ is congruent to $1$ modulo $4$.
2018 Belarusian National Olympiad
The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal.

Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.
2018 Belarusian National Olympiad
The point $X$ is marked inside the triangle $ABC$. The circumcircles of the triangles $AXB$ and $AXC$ intersect the side $BC$ again at $D$ and $E$ respectively. The line $DX$ intersects the side $AC$ at $K$, and the line $EX$ intersects the side $AB$ at $L$.

Prove that $LK\parallel BC$.
2018 Belarusian National Olympiad
Consider the expression $M(n, m)=|n\sqrt{n^2+a}-bm|$, where $n$ and $m$ are arbitrary positive integers and the numbers $a$ and $b$ are fixed, moreover $a$ is an odd positive integer and $b$ is a rational number with an odd denominator of its representation as an irreducible fraction.

Prove that there is

a) no more than a finite number of pairs $(n, m)$ for which $M(n, m)=0$;

b) a positive constant $C$ such that the inequality $M(n, m)\geqslant0$ holds for all pairs $(n, m)$ with $M(n, m)\ne 0$.
2018 Belarusian National Olympiad
The vertices of the regular $n$-gon are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center of the $n$-gon. The winner is a player if after his move it is possible to get any vertex from any other vertex moving along segments.

For each integer $n\geqslant 3$ determine who has a winning strategy.
2019 Belarusian National Olympiad
Is it true that for any nonzero rational numbers $a$ and $b$ one can find integers $m$ and $n$ such that the number $(am+b)^2+(a+nb)^2$ is an integer?



(M. Karpuk)
2019 Belarusian National Olympiad
The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$.

Find the value of angle $\angle AFB$.



(S. Mazanik)
2019 Belarusian National Olympiad
Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots.

Prove that $a-b>1$.



(V. Karamzin)
2019 Belarusian National Olympiad
The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$.

Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$.



(I. Voronovich)
2019 Belarusian National Olympiad
For a positive integer $n$ write down all its positive divisors in increasing order: $1=d_1<d_2<\ldots<d_k=n$.

Find all positive integers $n$ divisible by $2019$ such that $n=d_{19}\cdot d_{20}$.



(I. Gorodnin)
2019 Belarusian National Olympiad
The point $M$ is the midpoint of the side $BC$ of triangle $ABC$. A circle is passing through $B$, is tangent to the line $AM$ at $M$, and intersects the segment $AB$ secondary at the point $P$.

Prove that the circle, passing through $A$, $P$, and the midpoint of the segment $AM$, is tangent to the line $AC$.



(A. Voidelevich)
2019 Belarusian National Olympiad
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients such that $P(Q(x)^2)=P(x)\cdot Q(x)^2$.



(I. Voronovich)
2019 Belarusian National Olympiad
Andrey and Sasha play the game, making moves alternate. On his turn, Andrey marks on the plane an arbitrary point that has not yet been marked. After that, Sasha colors this point in one of two colors: white and black. Sasha wins if after his move it is impossible to draw a line such that all white points lie in one half-plane, while all black points lie in another half-plane with respect to this line.

a) Prove that Andrey can make moves in such a way that Sasha will never win.

b) Suppose that Andrey can mark only integer points on the Cartesian plane. Can Sasha guarantee himself a win regardless of Andrey's moves?



(N. Naradzetski)
2019 Belarusian National Olympiad
The two lines with slopes $2$ and $1/2$ pass through an arbitrary point $T$ on the axis $Oy$ and intersect the hyperbola $y=1/x$ at two points.

a) Prove that these four points lie on a circle.

b) The point $T$ runs through the entire $y$-axis. Find the locus of centers of such circles.



(I. Gorodnin)
2019 Belarusian National Olympiad
A point $P$ is chosen in the interior of the side $BC$ of triangle $ABC$. The points $D$ and $C$ are symmetric to $P$ with respect to the vertices $B$ and $C$, respectively. The circumcircles of the triangles $ABE$ and $ACD$ intersect at the points $A$ and $X$. The ray $AB$ intersects the segment $XD$ at the point $C_1$ and the ray $AC$ intersects the segment $XE$ at the point $B_1$.

Prove that the lines $BC$ and $B_1C_1$ are parallel.



(A. Voidelevich)
2019 Belarusian National Olympiad
The polynomial of seven variables

$$

Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)

$$is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:

$$

Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.

$$Find all possible values of $P_1(1,1,\ldots,1)$.



(A. Yuran)
2019 Belarusian National Olympiad
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.

Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums not greater than $9$.



(I. Gorodnin)
2019 Belarusian National Olympiad
Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$.



(I. Voronovich)
2019 Belarusian National Olympiad
The tangents to the circumcircle of the acute triangle $ABC$, passing through $B$ and $C$, meet at point $F$. The points $M$, $L$, and $N$ are the feet of perpendiculars from the vertex $A$ to the lines $FB$, $FC$, and $BC$, respectively.

Prove the inequality $AM+AL\ge 2AN$.



(V. Karamzin)
2019 Belarusian National Olympiad
The numbers $S_1=2^2, S_2=2^4,\ldots, S_n=2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$ the points $A_1,A_2,\ldots,A_n$ are consistently marked along the axis $Ox$, and the points $C_1,C_2,\ldots,C_n$ are consistently marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k=x_k$ and $C_{k-1}C_k=y_k$ are positive integers (let $A_0=C_0=O$, $A_n=A$, and $C_n=C$) and $x_k\cdot y_k=S_k$.

a) Find the maximal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this maximal area is achieved.

b) Find the minimal possible value of the area of the rctangle $OABC$ and all pairs of sets $(x_1,x_2,\ldots,x_k)$ and $(y_1,y_2,\ldots,y_k)$ at which this minimal area is achieved.



(E. Manzhulina, B. Rublyov)
2019 Belarusian National Olympiad
Call a polygon on a Cartesian plane to beinteger if all its vertices are integer. A convex integer $14$-gon is cut into integer parallelograms with areas not greater than $C$.

Find the minimal possible $C$.



(A. Yuran)
2019 Belarusian National Olympiad
a) Find all real numbers $a$ such that the parabola $y=x^2-a$ and the hyperbola $y=1/x$ intersect each other in three different points.

b) Find the locus of centers of circumcircles of such triples of intersection points when $a$ takes all possible values.



(I. Gorodnin)
2019 Belarusian National Olympiad
The polynomial

$$

Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2

$$is represented as the sum of squares of four polynomials of four variables with integer coefficients.

a) Find at least one such representation

b) Prove that for any such representation at least one of the four polynomials isidentically zero.



(A. Yuran)
2019 Belarusian National Olympiad
The sum of several (not necessarily different) real numbers from $[0,1]$ doesn't exceed $S$.

Find the maximum value of $S$ such that it is always possible to partition these numbers into two groups with sums $A\le 8$ and $B\le 4$.



(I. Gorodnin)
2019 Belarusian National Olympiad
The altitudes $CC_1$ and $BB_1$ are drawn in the acute triangle $ABC$. The bisectors of angles $\angle BB_1C$ and $\angle CC_1B$ intersect the line $BC$ at points $D$ and $E$, respectively, and meet each other at point $X$.

Prove that the intersection points of circumcircles of the triangles $BEX$ and $CDX$ lie on the line $AX$.



(A. Voidelevich)
2019 Belarusian National Olympiad
$n\ge 2$ positive integers are written on the blackboard. A move consists of three steps: 1) choose an arbitrary number $a$ on the blackboard, 2) calculate the least common multiple $N$ of all numbers written on the blackboard, and 3) replace $a$ by $N/a$.

Prove that using such moves it is always possible to make all the numbers on the blackboard equal to $1$.



(A. Naradzetski)
2019 Belarusian National Olympiad
The diagonals of the inscribed quadrilateral $ABCD$ intersect at the point $O$. The points $P$, $Q$, $R$, and $S$ are the feet of the perpendiculars from $O$ to the sides $AB$, $BC$, $CD$, and $DA$, respectively.

Prove the inequality $BD\ge SP+QR$.



(A. Naradzetski)
2019 Belarusian National Olympiad
Find all functions $f:\mathbb R\to\mathbb R$ satisfying the equality

$$

f(f(x)+f(y))=(x+y)f(x+y)

$$for all real $x$ and $y$.



(B. Serankou)
2019 Belarusian National Olympiad
At each node of the checkboard $n\times n$ board, a beetle sat. At midnight, each beetle crawled into the center of a cell. It turned out that the distance between any two beetles sitting in the adjacent (along the side) nodes didn't increase.

Prove that at least one beetle crawled into the center of a cell at the vertex of which it sat initially.



(A. Voidelevich)
2018 Bosnia And Herzegovina - Regional Olympiad
if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality:

$\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$
2018 Bosnia And Herzegovina - Regional Olympiad
Determine all triplets $(a,b,c)$ of real numbers such that sets $\{a^2-4c, b^2-2a, c^2-2b \}$ and $\{a-c,b-4c,a+b\}$ are equal and $2a+2b+6=5c$. In every set all elements are pairwise distinct
2018 Bosnia And Herzegovina - Regional Olympiad
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
2018 Bosnia And Herzegovina - Regional Olympiad
Prove that among arbitrary $13$ points in plane with coordinates as  integers, such that no three are collinear, we can pick three points as vertices of triangle such that its centroid has coordinates as integers.
2018 Bosnia And Herzegovina - Regional Olympiad
Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear
2018 Bosnia And Herzegovina - Regional Olympiad
Show that system of equations

$2ab=6(a+b)-13$

$a^2+b^2=4$

has not solutions in set of real numbers.
2018 Bosnia And Herzegovina - Regional Olympiad
Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer
2018 Bosnia And Herzegovina - Regional Olympiad
Solve equation $x \lfloor{x}\rfloor+\{x\}=2018$, where $x$ is real number
2018 Bosnia And Herzegovina - Regional Olympiad
Let $P$ be a point on circumcircle of triangle $ABC$ on arc  $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$  intersect at point $E$, and lines $AC$ and $BP$ intersect  at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that:

${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$
2018 Bosnia And Herzegovina - Regional Olympiad
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If $W$ is number of remaining white chips, and $B$ number of remaining black chips on board and $A=min\{W,B\}$, determine maximum of $A$
2018 Bosnia And Herzegovina - Regional Olympiad
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions
2018 Bosnia And Herzegovina - Regional Olympiad
Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum

$$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$can be negative and can also be positive. Find the minimal value of this sum
2018 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ given is point $P$ such that $\angle ACP = \angle ABP = 10^{\circ}$, $\angle CAP = 20^{\circ}$ and

$\angle BAP = 30^{\circ}$. Prove that $AC=BC$
2018 Bosnia And Herzegovina - Regional Olympiad
We observe that number $10001=73\cdot137$ is not prime. Show that every member of infinite sequence $10001, 100010001, 1000100010001,...$ is not prime
2018 Bosnia And Herzegovina - Regional Olympiad
It is given $2018$ points in plane. Prove that it is possible to cover them with circles such that:

$i)$ sum of lengths of all diameters  of all circles is not greater than $2018$

$ii)$ distance between any two circles is greater than $1$
2018 Bosnia And Herzegovina - Regional Olympiad
$a)$ Prove that for all positive integers $n \geq 3$ holds:

$$\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2$$where $\binom{n}{k}$ , with integer $k$ such that $n \geq k \geq 0$, is binomial coefficent



$b)$ Let $n \geq 3$ be an odd positive integer. Prove that set $A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}}    \right\}$ has odd number of odd numbers
2018 Bosnia And Herzegovina - Regional Olympiad
If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality:

$\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$
2018 Bosnia And Herzegovina - Regional Olympiad
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$
2017 Bosnia And Herzegovina - Regional Olympiad
Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$

Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$
2017 Bosnia And Herzegovina - Regional Olympiad
Prove that numbers $1,2,...,16$ can be divided in sequence such that sum of any two neighboring numbers is perfect square
2017 Bosnia And Herzegovina - Regional Olympiad
Does there exist positive integer $n$  such that sum of all digits of number $n(4n+1)$ is equal to $2017$
2017 Bosnia And Herzegovina - Regional Olympiad
It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$
2017 Bosnia And Herzegovina - Regional Olympiad
If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ ,  are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$
2017 Bosnia And Herzegovina - Regional Olympiad
It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$
2017 Bosnia And Herzegovina - Regional Olympiad
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares
2017 Bosnia And Herzegovina - Regional Olympiad
Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$
2017 Bosnia And Herzegovina - Regional Olympiad
In terms of real parameter $a$ solve inequality:

$\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$

in set of real numbers
2017 Bosnia And Herzegovina - Regional Olympiad
Let $ABC$ be an isosceles triangle such that $AB=AC$. Find angles of triangle $ABC$ if

$\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}$
2017 Bosnia And Herzegovina - Regional Olympiad
Let $S$ be a set of $6$ positive real numbers such that

$\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$

Prove that if we sort these numbers in ascending order, then they form an arithmetic progression
2017 Bosnia And Herzegovina - Regional Olympiad
It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$
2017 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ on side $AC$ are points $K$, $L$ and $M$ such that $BK$ is an angle bisector of $\angle ABL$, $BL$ is an angle bisector of $\angle KBM$ and $BM$ is an angle bisector of $\angle LBC$, respectively. Prove that $4 \cdot LM <AC$ and $3\cdot \angle BAC - \angle ACB < 180^{\circ}$
2017 Bosnia And Herzegovina - Regional Olympiad
How many knights you can put on chess table $5 \times 5$ such that every one of them attacks exactly two other knights ?
2016 Bosnia And Herzegovina - Regional Olympiad
Find minimal value of $A=\frac{\left(x+\frac{1}{x}\right)^6-\left(x^6+\frac{1}{x^6}\right)-2}{\left(x+\frac{1}{x}\right)^3+\left(x^3+\frac{1}{x^3}\right)}$
2016 Bosnia And Herzegovina - Regional Olympiad
Let $ABC$ be an isosceles triangle  such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$
2016 Bosnia And Herzegovina - Regional Olympiad
Nine lines are given such that every one of them intersects given square $ABCD$ on two trapezoids, which area ratio is $2 : 3$. Prove that at least $3$ of those $9$ lines pass through the same point
2016 Bosnia And Herzegovina - Regional Olympiad
Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$
2016 Bosnia And Herzegovina - Regional Olympiad
If $\mid ax^2+bx+c \mid \leq 1$ for all $x \in [-1,1]$ prove that:

$a)$ $\mid c \mid \leq 1$

$b)$ $\mid a+c \mid \leq 1$

$c)$ $a^2+b^2+c^2 \leq 5$
2016 Bosnia And Herzegovina - Regional Olympiad
Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square
2016 Bosnia And Herzegovina - Regional Olympiad
Let $AB$ be a diameter of semicircle $h$. On this semicircle there is point $C$, distinct from points $A$ and $B$. Foot of perpendicular from point $C$ to side $AB$  is point $D$. Circle $k$ is outside the triangle $ADC$ and at the same time touches semicircle $h$ and sides $AB$ and $CD$. Touching point of $k$ with side $AB$ is point $E$, with semicircle $h$ is point $T$ and with side $CD$ is point $S$

$a)$ Prove that points $A$, $S$ and $T$ are collinear

$b)$ Prove that $AC=AE$
2016 Bosnia And Herzegovina - Regional Olympiad
Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$
2016 Bosnia And Herzegovina - Regional Olympiad
Let $a$ and $b$ be real numbers bigger than $1$. Find maximal value of $c \in \mathbb{R}$ such that $$\frac{1}{3+\log _{a} b}+\frac{1}{3+\log _{b} a} \geq c$$
2016 Bosnia And Herzegovina - Regional Olympiad
Does there exist a right angled triangle, which hypotenuse is $2016^{2017}$ and two other sides positive integers.