BALKAN-SL 2015 N5

For a positive integer $s$ , denote with $v_2(s)$ the maximum power of $2$ that divides $s$ . Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)