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(2^n -1)!! -1 divides 2^n

Source: 2023 Grand Duchy of Lithuania, MC p4 (Baltic Way TST)

March 23, 2024
number theorydividesfactorial

Problem Statement

Note that k≥1k\ge 1 for an odd natural number k!!=k⋅(k−2)⋅...⋅1.k! ! = k \cdot (k - 2) \cdot ... \cdot 1. Prove that (2n−1)!!−1(2^n -1)!! -1 divides 2n2^n for all n≥3n \ge 3.
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