Câu hỏi 82
Chứng minh với $a,b,c$ dương và $l\gt m$,
$$\begin{aligned}&\frac{a^{3l}+a^{3m}+1}{a^l+b^{l-m}a^m+b^l}\\&+\frac{b^{3l}+b^{3m}+1}{b^l+c^{l-m}b^m+c^l}\\&+\frac{c^{3l}+c^{3m}+1}{c^l+a^{l-m}c^m+a^l}\ge a^m+b^m+c^m.\end{aligned}$$
$$\begin{aligned}&\frac{a^{3l}+a^{3m}+1}{a^l+b^{l-m}a^m+b^l}\\&+\frac{b^{3l}+b^{3m}+1}{b^l+c^{l-m}b^m+c^l}\\&+\frac{c^{3l}+c^{3m}+1}{c^l+a^{l-m}c^m+a^l}\ge a^m+b^m+c^m.\end{aligned}$$