Question 1
Fix an integer $n \ge 2$.
For which real numbers $x$ is
$$\lfloor nx\rfloor-\sum_{k=1}^n\frac{\lfloor kx\rfloor}{k}$$
maximal, and what is the maximal value that this expression can take?
Proposed by Titu Andreescu and Gabriel Dospinescu.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
For which real numbers $x$ is
$$\lfloor nx\rfloor-\sum_{k=1}^n\frac{\lfloor kx\rfloor}{k}$$
maximal, and what is the maximal value that this expression can take?
Proposed by Titu Andreescu and Gabriel Dospinescu.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.