## [Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2015

### Algebra

1. Let $x$, $y$, $z$ be real numbers, satisfying the relations $x \ge 20$, $y \ge 40$, $z \ge 1675$ and $x + y + z = 2015$. Find the greatest value of the product $P = xy z$
2. Assume that $x$ satisfies $$x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0.$$ Find the value of $$x^6-8x^5+13x^4-5x^3+49x^2-137x+2015.$$
3. Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{b}+\sqrt{\frac{b}{c}}+\sqrt{\frac{c}{a}}>2.$$
4. Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression $A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.$
5. The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that $$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

### Geometry

1. Around the triangle $ABC$ the circle is circumscribed, and at the vertex ${C}$ tangent ${t}$ to this circle is drawn. The line ${p}$, which is parallel to this tangent intersects the lines ${BC}$ and ${AC}$ at the points ${D}$ and ${E}$, respectively. Prove that the points $A$, $B$, $D$, $E$ belong to the same circle.
2. The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
3. Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$, say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$, say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$.
4. Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively. Prove that if $D$ is the intersection point of the lines $EF$ and $MC$ then $\angle ADB = \angle EMF.$
5. Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle κύκλος touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$onto ${BC}$ meets ${EF}$at ${M}$, and similarly the perpendicular line erected at ${B}$onto ${BC}$ meets ${EF}$at${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$.

### Number Theory

1. What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?.
2. A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$. Prove that a) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$. b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$.
3. a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$. b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$.
4. Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation $a^2+b^2+16c^2 = 9k^2 + 1.$
5. Check if there exists positive integers $a, b$ and prime number $p$ such that $$a^3-b^3=4p^2$$

### Combinatorics

1. A board $n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.
2. $2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.
3. Positive integers are put into the following table $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{array}$$
Find the number of the line and column where the number $2015$ stays.
4. Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$.
5. An $L$-shape is one of the following four pieces, each consisting of three unit squares. A $5\times 5$ board, consisting of $25$ unit squares, a positive integer $k\leq 25$ and an unlimited supply of L-shapes are given. Two players $A$ and $B$, play the following game: starting with $A$ they play alternatively mark a previously unmarked unit square until they marked a total of $k$ unit squares. e say that a placement of $L$-shapes on unmarked unit squares is called good if the L-shapes do not overlap and each of them covers exactly three unmarked unit squares of the board. $B$ wins if every good placement of $L$-shapes leaves uncovered at least three unmarked unit squares. Determine the minimum value of $k$ for which $B$ has a winning strategy.