### International Mathematics Competition for University Students

July 27 - Aug 2 2015, Blagoevgrad, Bulgaria

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### Problem 2

2. For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that $\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.$ When does equality hold?

Proposed by Stephan Wagner, Stellenbosch University

Hint: Determine $f(k)$ between two consective powers of $2$.