## [Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2012

### Algebra

1. Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).$ When does equality hold?
2. Let $a$, $b$, $c$ be positive real numbers such that $abc=1$ . Show that $\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}$
3. Let $a$, $b$, $c$ be positive real numbers such that $$a+b+c=a^2+b^2+c^2.$$ Prove that $\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}$
4. Solve the following equation for $x , y , z \in \mathbb{N}$ $\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}$
5. Find the largest positive integer $n$ for which the inequality $\frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}$ holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt{abc}=abc$.

### Geometry

1. Let $ABC$ be an equilateral triangle and $P$ be a point on the circumcircle of the triangle but distinct from $A$, $B$ and $C$. The lines through $P$ and parallel to $BC$, $CA$, $AB$ intersect the lines $CA$, $AB$, $BC$ at $M$, $N$ and $Q$ respectively. Prove that $M$, $N$ and $Q$ are collinear .
2. Let $ABC$ be an isosceles triangle with $AB=AC$. Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$. Prove that $HK \bot AB$.
3. Let $AB$ and $CD$ be chords in a circle of center $O$ with $A$, $B$, $C$, $D$ distinct, and with the lines $AB$ and $CD$ meeting at a right angle at point $E$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively. Prove that if $MN \bot OE$ then $AD \parallel BC$.
4. Let $ABC$ be an acute-angled triangle with circumcircle $\omega$, and let $O$, $H$ be the triangle's circumcenter and orthocenter respectively. Let also $A'$ be the point where the angle bisector of the angle $BAC$ meets $\omega$. Find the measure of the angle $BAC$ if $A'H=AH$.
5. Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.
6. Let $O_1$ be a point in the exterior of the circle $\omega$ of center $O$ and radius $R$, and let $O_1N$, $O_1D$ be the tangent segments from $O_1$ to the circle. On the segment $O_1N$ consider the point $B$ such that $BN=R$. Let the line from $B$ parallel to $ON$ intersect the segment $O_1D$ at $C$. If $A$ is a point on the segment $O_1D$ other than $C$ so that $BC=BA=a$, and if the incircle of the triangle $ABC$ has radius $r$, then find the area of $\triangle ABC$ in terms of $a$, $R$, $r$.
7. Let $MNPQ$ be a square of side length $1$, and $A$, $B$, $C$, $D$ points on the sides $MN$, $NP$, $PQ$ and $QM$ respectively such that $AC \cdot BD=\dfrac{5}{4}$. Can the set $\{AB , BC , CD , DA \}$ be partitioned into two subsets $S_1$ and $S_2$ of two elements each so that each one has the sum of his elements a positive integer?.

### Combinatorics

1. Along a round table are arranged $11$ cards with the names ( all distinct ) of the $11$ members of the $16^{th}$ JBMO Problem Selection Committee. The cards are arranged in a regular polygon manner. Assume that in the first meeting of the Committee none of its $11$ members sits in front of the card with his name. Is it possible to rotate the table by some angle so that at the end at least two members sit in front of the card with their names?.
2. On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors. a) Can $n$ be $6$?. b) Can $n$ be $7$?.
3. In a circle of diameter $1$ consider $65$ points, no three of them collinear. Prove that there exist three among these points which are the vertices of a triangle with area less than or equal to $\dfrac{1}{72}$.

### Number Theory

1. let $a$, $b$ be integers and $$s=a^3+b^3-60ab(a+b)\geq 2012.$$ Find the least possible value of $s$.
2. Do there exist prime numbers $p$ and $q$ such that $$p^2(p^3-1)=q(q+1)$$
3. Decipher the equality $(\overline{VER}-\overline{IA})=G^{R^E} (\overline {GRE}+\overline{ECE})$ assuming that the number $\overline {GREECE}$ has a maximum value. Each letter corresponds to a unique digit from $0$ to $9$ and different letters correspond to different digits. It's also supposed that all the letters $G$, $E$, $V$ and $I$ are different from $0$.
4. Determine all triples $(m , n , p)$ satisfying $n^{2p}=m^2+n^2+p+1$ where $m$ and $n$ are integers and $p$ is a prime number.
5. Find all positive integers $x,y,z$ and $t$ such that $$2^x3^y+5^z=7^t.$$
6. Let $a$, $b$, $c$, $d$ be integers and \begin{align*}A&=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4,\\ B&=d(d+1)(d+2)(d+3)+1.\end{align*} Prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.
7. Find all $a , b , c \in \mathbb{N}$ for which $1997^a+15^b=2012^c$