Let $f(x)$ be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers $n$, for which $f(n)$ is not divisible by any of $f(2)$, $f(3)$, ..., $f(n-1)$.

Let $n\geq 2$ be a positive integer. We call a vertex every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an edge, if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is vital for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.

Given is a convex cyclic pentagon $ABCDE$ and a point $P$ inside it, such that $AB=AE=AP$ and $BC=CE$. The lines $AD$ and $BE$ intersect in $Q$. Points $R$ and $S$ are on segments $CP$ and $BP$ such that $DR=QR$ and $SR||BC$. Show that the circumcircles of $BEP$ and $PQS$ are tangent to each other.