Problem

Source: 2019 Taiwan TST Round 1

Tags: factorial, number theory



Given a positive integer $ n $, let $ A, B $ be two co-prime positive integers such that $$ \frac{B}{A} = \left(\frac{n\left(n+1\right)}{2}\right)!\cdot\prod\limits_{k=1}^{n}{\frac{k!}{\left(2k\right)!}} $$Prove that $ A $ is a power of $ 2 $.