Prove that for any positive integers $a_1, a_2, \ldots , a_n$ the following inequality holds true:
\[\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\geqslant a_1+a_2+\cdots+a_n.\]Maxim Didin
Applying AM-GM, we get$$\frac{a_i^2}{a_{i+1}}+a_{i+1}\ge 2a_i$$$\Rightarrow \frac{a_i^2}{a_{i+1}}\ge 2a_i-a_{i+1}$
As $2a_i-a_{i+1}$ is an integer, so$$\left\lfloor\frac{a_i^2}{a_{i+1}}\right\rfloor\ge 2a_i-a_{i+1}$$$\Rightarrow \sum_{i=1}^{n}\left\lfloor\frac{a_i^2}{a_{i+1}}\right\rfloor\ge a_1+a_2+\cdots+a_n$.$\quad\square$