(2017 Tsinghua University THUSSAT)
Denote the largest integer not exceeding the real number $x$ as $[x]$, take any positive odd number $m,n$ that is relatively prime and not less than 3, let
$$I=\sum_{i=1}^{\frac {m-1}{2}}\left[\frac{ni}{m}\right]+
\sum_{j=1}^{\frac{n-1}{2}}\left[\frac{ mi}{n}\right],$$
then ( )
A.$I<\dfrac{m-1}{2}\cdot\dfrac{n-1}{2}$
B.$I>\dfrac{ m-1}{2}\cdot\dfrac{n-1}{2}$
C.$I\leq\dfrac{m-1}{2}\cdot\dfrac{n-1}{2}$
D .$I\geq\dfrac{m-1}{2}\cdot\dfrac{n-1}{2}$
Answer: CD
Hint: $I=\dfrac{m-1}{2}\cdot\dfrac{n-1}{2}$, see Min Sihe's Theory of Elementary Numbers, Chapter 5, Section 4, Quadratic Reciprocity Theorem Prove some of the content.