1)In a room let there be $\displaystyle n>1$ people. Show that there must exist two people that shaked the same number of hands with others.

2)Let $\displaystyle a_1,a_2,...,a_n$ be distinct integers from $\displaystyle \{1,2,...,n\}$ not necessarily in that order. Show that if $\displaystyle n$ is odd then $\displaystyle (a_1-1)(a_2-2)...(a_n-n)$ is an even number.