CSMO Day 2 P2

Inductive Method. Use local Inq. x^2+2/x≥3, namely x^3-3x+2≥0, or (x-1)^2*(x+2)≥0.

sqing wrote: Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$ \left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$

Attachments:

Prolety-Maderusat wrote: Inductive Method. Use local Inq. $x^2+\frac{2}{x}\ge 3$, namely $x^3-3x+2\ge 0$, or $(x-1)^2\cdot (x+2)\ge 0.$