CMO 2013 Problem 4

Let $n$ be a positive integer. For any positive integer $j$ and positive real number $r$, define

$$f_{j}(r)=\min (j r, n)+\min \left(\frac{j}{r}, n\right), \text { and } g_{j}(r)=\min (\lceil j r\rceil, n)+\min \left(\left\lceil\frac{j}{r}\right\rceil, n\right) \text {, }$$

where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Prove that

$$\sum_{j=1}^{n} f_{j}(r) \leq n^{2}+n \leq \sum_{j=1}^{n} g_{j}(r)$$