IMC 2015 Problem 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)