[warutT]

ข้อ 34 ครับ

From Cauchy-Schwarz Inequality

$ab+bc+ca \leq \sqrt{a^2+b^2+c^2}\sqrt{a^2+b^2+c^2}=a^2+b^2+c^2 ...(*)$

From modified Cauchy-Schwarz Inequality

$\dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}$

$=\dfrac{a^2}{ab+2ac}+\dfrac{b^2}{bc+2ab}+\dfrac{c^2}{ac+2bc}$

$\geq\dfrac{(a+b+c)^2}{ab+2ac+bc+2ab+ac+2bc}$

$=\dfrac{(a+b+c)^2}{ab+bc+ac+2ab+2bc+2ac}$

$\geq \dfrac{(a+b+c)^2}{a^2+b^2+c^2+2ab+2bc+2ac} ...(From (*))$

$=\dfrac{(a+b+c)^2}{(a+b+c)^2}=1$

$\therefore \dfrac{a}{b+2c}+\dfrac{b}{c+2a}+\dfrac{c}{a+2b}\geq 1$

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