The natural numbers $a$ and $b$ satisfy the inequalities $a > b > 1$ . It is also known that the equation $\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$. Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).

In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.

Through the points with integer coordinates in the right-angled coordinate system $Oxyz$ are constructed planes, parallel to the coordinate planes and in this way the space is divided to unit cubes. Find all triples ($a, b, c$) consisting of natural numbers ($a \le b \le c$) for which the cubes formed can be coloured in $abc$ colors in such a way that every palellepiped with dimensions $a \times b \times c$, having vertices with integer coordinates and sides parallel to the coordinate axis doesn't contain unit cubes in the same color.

Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which \[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\] for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).

We divide a convex $2009$-gon in triangles using non-intersecting diagonals. One of these diagonals is colored green. It is allowed the following operation: for two triangles $ABC$ and $BCD$ from the dividing/separating with a common side $BC$ if the replaced diagonal was green it loses its color and the replacing diagonal becomes green colored. Prove that if we choose any diagonal in advance it can be colored in green after applying the operation described finite number of times.

Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$ are positive real numbers, than $ \left(\sum_{i,j = 1}^{n}\frac {a_{i}a_{j}}{c_{i} + c_{j}}\right)\left(\sum_{i,j = 1}^{n}\frac {b_{i}b_{j}}{c_{i} + c_{j}}\right)\ge \left(\sum_{i,j = 1}^{n}\frac {a_{i}b_{j}}{c_{i} + c_{j}}\right)^{2}$.