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?

Let $a$ , $b$ , $c$ be positive real numbers such that $abc=1$ . Show that : \[\frac{1}{a^3+bc}+\frac{1}{b^3+ca}+\frac{1}{c^3+ab} \leq \frac{ \left (ab+bc+ca \right )^2 }{6}\]

Let $a$ , $b$ , $c$ be positive real numbers such that $a+b+c=a^2+b^2+c^2$ . Prove that : \[\frac{a^2}{a^2+ab}+\frac{b^2}{b^2+bc}+\frac{c^2}{c^2+ca} \geq \frac{a+b+c}{2}\]

Solve the following equation for $x , y , z \in \mathbb{N}$ : \[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]

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 $\sqrt[1]{abc}=abc$.

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 .

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 $.

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 . If $MN \bot OE$ , prove that $AD \parallel BC$.

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 $\omega$ . If $A^{'}H=AH$ , then find the measure of the angle $BAC$.

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 $NMB$.

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 $a ,R ,r$.

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 $AC \cdot BD=\frac{5}{4}$. 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?

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 ?

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 $7$ ?

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 $\frac{1}{72}$.

If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$.

Do there exist prime numbers $p$ and $q$ such that $p^2(p^3-1)=q(q+1)$ ?

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 $0$.

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.

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

If $a$ , $b$ , $c$ , $d$ are integers and $A=2(a-2b+c)^4+2(b-2c+a)^4+2(c-2a+b)^4$ , $B=d(d+1)(d+2)(d+3)+1$ , then prove that $\left (\sqrt{A}+1 \right )^2 +B$ cannot be a perfect square.

Find all $a , b , c \in \mathbb{N}$ for which \[1997^a+15^b=2012^c\]