Let $ X$, $ Y$, $ Z$ be distinct positive integers having exactly two digits in such a way that: $ X=10a +b$ $ Y=10b +c$ $ Z=10c +a$ ($ a,b,c$ are digits) Find all posible values of $ gcd(X,Y,Z)$

Let $ ABCD$ be a trapezium of parallel sides $ AD$ and $ BC$ and non-parallel sides $ AB$ and $ CD$ Let $ I$ be the incenter of $ ABC$. It is known that exists a point $ Q \in AD$ with $ Q \neq A$ and $ Q \neq D$ such that if $ P$ is a point of the intersection of the bisectors of $ \widehat{ CQD}$ and $ \widehat{CAD}$ then $ PI \parallel AD$ Prove that $ PI=BQ$

A $3000\times 3000$ square is tiled by dominoes (i. e. $1\times 2$ rectangles) in an arbitrary way. Show that one can color the dominoes in three colors such that the number of the dominoes of each color is the same, and each dominoe $d$ has at most two neighbours of the same color as $d$. (Two dominoes are said to be neighbours if a cell of one domino has a common edge with a cell of the other one.)

Find all real values of $ x>1$ which satisfy: $ \frac{x^2}{x-1} + \sqrt{x-1} +\frac{\sqrt{x-1}}{x^2} = \frac{x-1}{x^2} + \frac{1}{\sqrt{x-1}} + \frac{x^2}{\sqrt{x-1}}$

Let $ d_1,d_2 \ldots, d_r$ be the positive divisors of $ n$ $ 1=d_1<d_2< \ldots <d_r=n$ If $ (d_7)^2 + (d_{15})^2= (d_{16})^2$ find all posible values of $ d_{17}$

For natural $ n$ we define $ s(n)$ as the sum of digits of $ n$ (in base ten) Does there exist a positive real constant $ c$ such that for all natural $ n$ we have $ \frac{s(n)}{s(n^2)} \le c$ ?