## [Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2017

### Algebra

1. Let $a, b, c$ be positive real numbers such that $$a + b + c + ab + bc + ca + abc = 7.$$ Prove that $$\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6.$$
2. Let $a$ and $b$ be positive real numbers such that $$3a^2 + 2b^2 = 3a + 2b.$$ Find the minimum value of $$A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}}$$
3. Let $a\le b\le c \le d$. Show that $$ab^3+bc^3+cd^3+da^3\ge a^2b^2+b^2c^2+c^2d^2+d^2a^2$$
4. Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$. Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$ When does the equality hold?

### Combinatorics

1. Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.
2. Consider a regular $2n$-gon $P$, $A_1,A_2,\cdots ,A_{2n}$ in the plane where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We color the sides of $P$ in $3$ different colors (ignore the vertices of $P$, we consider them colorless), such that every side is colored in exactly one color, and each color is used at least once. Moreover ,from every point in the plane external to $P$, points of most $2$ different colors on $P$ can be seen .Find the number of distinct such colorings of $P$ (two colorings are considered distinct if at least one of sides is colored differently).
3. We have two piles with $2000$ and $2017$ coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile $t$ coins for some $2\le t \le 4$, and adds to the other pile $1$ coin. The players can choose a different $t$ at each turn, and the player who cannot make a move loses. If Ann plays first determine which player has a winning strategy.

### Geometry

1. Given a parallelogram $ABCD$. The line perpendicular to $AC$ passing through $C$ and the line perpendicular to $BD$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $PC$ intersects the line $BC$ at point $X$, ($X \ne C$) and the line $DC$ at point $Y$ ($Y \ne C$). Prove that the line $AX$ passes through the point $Y$ .
2. Let $ABC$ be an acute triangle such that $AB$ is the shortest side of the triangle. Let $D$ be the midpoint of the side $AB$ and $P$ be an interior point of the triangle such that $\angle CAP = \angle CBP = \angle ACB$. Denote by M and $N$ the feet of the perpendiculars from $P$ to $BC$ and $AC$, respectively. Let $p$ be the line through $M$ parallel to $AC$ and $q$ be the line through $N$ parallel to $BC$. If $p$ and $q$ intersect at $K$ prove that $D$ is the circumcenter of triangle $MNK$.
3. Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$. Let $S$ be the midpoint of segment $AD$. Prove that if $\angle ADE = \angle ABC - \angle ACB$ then $\angle BSC = 2 \angle BAC$.
4. Let $ABC$ be an acute triangle such that $AB\neq AC$ with circumcircle $\Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $\Gamma$ such that $AD \perp BC$. Let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $\Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $\Gamma$ at point $X\neq E$. Prove that the point $A$, $M$ and $X$ are collinear.
5. A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE$, $BCEF$, $CAFD$, $AEPF$, $BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

### Number Theory

1. Determine all the sets of six consecutive positive integers such that the product of some two of them added to the product of some other two of them is equal to the product of the remaining two numbers.
2. Determine all positive integers $n$ such that $$n^2 \mid (n - 1)!$$
3. Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.
4. Solve in nonnegative integers the equation $$5^t + 3^x4^y = z^2.$$
5. Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$. (A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.)