Áp dụng BĐT Cô-si:
\(a^2+b^2\ge2ab\Rightarrow\dfrac{a}{a^2+b^2}\le\dfrac{a}{2ab}=\dfrac{1}{2b}\)
Tương tự: \(\dfrac{b}{b^2+c^2}\le\dfrac{1}{2c}\); \(\dfrac{c}{c^2+a^2}\le\dfrac{1}{2a}\)
Cộng vế:
\(\dfrac{a}{a^2+b^2}+\dfrac{b}{b^2+c^2}+\dfrac{c}{c^2+a^2}\le\dfrac{1}{2}\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
Dấu "=" xảy ra khi \(a=b=c\)