DANUBE 2007

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 $\min_{1\le i\le n-1}\left|\sigma(i+1)-\sigma(i)\right|$ . Determine $\max_{\sigma\in S_n}l(\sigma)$ .

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 $i=\overline{1,\ldots,n}$ .