MathDB

v_2 (\prod_{n=1}^{2^m}\binom{2n}{n} )=m2^{m-1}+1

Source: Balkan BMO Shortlist 2015 N5

August 5, 2019

number theorymaximumpower of 2dividesProductIMO Shortlist

Problem Statement

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)