The points $A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}$ are on the sides $AB$, $BC$ and $AC$ of an acute triangle $ABC$ such that $AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB$, $BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC$ and $CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC$. Let $k_{A}, k_{B}$ and $k_{C}$ be the circumcircles of the triangles $AA_{1}C_{2}$, $BB_{1}A_{2}$ and $CC_{1}B_{2}$ respectively. Furthermore, let $a_{B}$ and $a_{C}$ be the tangents to $k_{A}$ at $A_{1}$ and $C_{2}$, $b_{C}$ and $b_{A}$ the tangents to $k_{B}$ at $B_{1}$ and $A_{2}$ and $c_{A}$ and $c_{B}$ the tangents to $k_{C}$ at $C_{1}$ and $B_{2}$. Show that the perpendicular lines from the intersection points of $a_{B}$ and $b_{A}$, $b_{C}$ and $c_{B}$, $c_{A}$ and $a_{C}$ to $AB$, $BC$ and $CA$ respectively are concurrent.

a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational. b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.

Denote by $\mathbb{Z}^{*}$ the set of all nonzero integers and denote by $\mathbb{N}_{0}$ the set of all nonnegative integers. Find all functions $f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}$ such that: $(1)$ For all $a,b \in \mathbb{Z}^{*}$ such that $a+b \in \mathbb{Z}^{*}$ we have $f(a+b) \geq $ min $\left \{ f(a),f(b) \right \}$. $(2)$ For all $a, b \in \mathbb{Z}^{*}$ we have $f(ab) = f(a)+f(b)$.

Let $a>0,b>0,c>0$ and $a+b+c=1$. Show the inequality $$\frac{a^4+b^4}{a^2+b^2}+\frac{b^3+c^3}{b+c} + \frac{2a^2+b^2+2c^2}{2} \geq \frac{1}{2}$$

Let $ABC$ be a triangle with given sides $a,b,c$. Determine the minimal possible length of the diagonal of an inscribed rectangle in this triangle. Note: A rectangle is inscribed in the triangle if two of its consecutive vertices lie on one side of the triangle, while the other two vertices lie on the other two sides of the triangle.

Let $a$ and $n>0$ be integers. Define $a_{n} = 1+a+a^2...+a^{n-1}$. Show that if $p|a^p-1$ for all prime divisors of $n_{2}-n_{1}$, then the number $\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}$ is an integer.