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Eventually integers will be observed

Source: Azerbaijan NMO 2016. Senior P4

July 27, 2023
number theorycombinatorics

Problem Statement

Let A=1⋅3⋅5⋅...⋅(2n−1)2⋅4⋅6⋅...⋅(2n)A = \frac{1 \cdot 3 \cdot 5\cdot ... \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot (2n)} Prove that in the infinite sequence A,2A,4A,8A,...,2kA,….A, 2A, 4A, 8A, ..., 2^k A, …. only integers will be observed, eventually.
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