Question 6
For a real number $x$, let $\lfloor x\rfloor$ be the greatest integer not exceeding $x$. Prove that for all positive integers $a_1,\ldots,a_n$,
$$\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\ge a_1+\cdots+a_n.$$
(9 points)
Neutral editorial illustration.
$$\left\lfloor\frac{a_1^2}{a_2}\right\rfloor+\left\lfloor\frac{a_2^2}{a_3}\right\rfloor+\cdots+\left\lfloor\frac{a_n^2}{a_1}\right\rfloor\ge a_1+\cdots+a_n.$$
(9 points)
Neutral editorial illustration.