Question 6
Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define
$$
f_{j}(r)=\min (j r, n)+\min \left(\frac{j}{r}, n\right), \quad \text{and} \quad g_{j}(r)=\min (\lceil j r\rceil, n)+\min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),
$$
where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that
$$
\sum_{j=1}^{n} f_{j}(r) \leq n^{2}+n \leq \sum_{j=1}^{n} g_{j}(r)
$$
$$
f_{j}(r)=\min (j r, n)+\min \left(\frac{j}{r}, n\right), \quad \text{and} \quad g_{j}(r)=\min (\lceil j r\rceil, n)+\min \left(\left\lceil\frac{j}{r}\right\rceil, n\right),
$$
where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that
$$
\sum_{j=1}^{n} f_{j}(r) \leq n^{2}+n \leq \sum_{j=1}^{n} g_{j}(r)
$$