Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n-1}\left|\sigma(i+1)-\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.

Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.

For each positive integer $ n$, define $ f(n)$ as the exponent of the $ 2$ in the decomposition in prime factors of the number $ n!$. Prove that the equation $ n-f(n)=a$ has infinitely many solutions for any positive integer $ a$.

Let $ a,n$ be positive integers such that $ a\ge(n-1)!$. Prove that there exist $ n$ distinct prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a+i$, for all $ i=\overline{1,\ldots,n}$.