Find all polynomials $P(x)$ such that for every real $x$ it hold $(x+100)P(x)-xP(x+1)=1$.

Let $ABCD$ be cyclic quadriateral and let $AC$ and $BD$ intersect at $E$ and $AB$ and $CD$ at $F$. Let $K$ be point in plane such that $ABKC$ is parallelogram. Prove $\angle AFE=\angle CDF$.

Define $corner $ as a 'broken' line(in Cartesian coordinate plane) consisting of one vertical and one horizontal line, with $ends$ at first point and last point of 'broken' line (for example $ABC$ is corner if $B$ is in plane such that $AB\perp BC$ and $AB||x$ or $AB||y$ ( note that in following statement one chooses one of such $B$)). In Cartesian coordinate plane there are $n$ blue and $n$ red points with all different $x$ and $y$ coordinates. Prove that one can draw $n$ $corners $ without common points such that every $corner $ has one blue and one red $end$.

Let {$a_n$}$_{1}^{\infty}$ be array such that $a_1=2$ and for every $n\ge1$ $a_{n+1}=2^{a_n}+2$. Let $m,n$ be positive integers such that $m<n$. Prove that $a_m|a_n$.