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Found problems: 26

2014 Canadian Mathematical Olympiad Qualification, 8
Tags: number theory , binary
For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.