MathDB

Let

n

be a positive integer. For any positive integer

j

and positive real number

r

, define

f_j(r)

and

g_j(r)

f_j(r) = \min (jr, n) + \min\left(\frac{j}{r}, n\right), \text{ and } g_j(r) = \min (\lceil jr\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)

for all positive real numbers

r

Problem Statement