2010 KMO #2

2. 양의 실수 a,b,cab+bc+ca=1을 만족할 때 다음 부등식이 성립함을 보여라.
\sqrt{a^2+b^2+\frac{1}{c^2}} + \sqrt{b^2+c^2+\frac{1}{a^2}} + \sqrt{c^2+a^2+\frac{1}{b^2}} \geq \sqrt{33}

코시-슈바르츠 부등식에 의해 \displaystyle \left( \frac{(ab+bc+ca)^2}{a^2} + b^2+c^2 \right)^2 (3^2+1^2+1^2) \geq \left( 3 \frac{ab+bc+ca}{a}+b+c \right)^2이므로 \displaystyle (LHS) \geq \sum \frac{1}{\sqrt{11}}(3 \frac{ab+bc+ca}{a}+b+c)
\displaystyle = \frac{1}{\sqrt{11}}(3\frac{(ab+bc+ca)^2}{abc} + 2(a+b+c))
\displaystyle \geq \frac{1}{\sqrt{11}} 11(a+b+c) = \sqrt{11}(a+b+c) = \sqrt{11(a+b+c)^2}
\displaystyle \geq \sqrt{33(ab+bc+ca)} = \sqrt{33}, 증명끝.

등호는 a=b=c=\frac{1}{\sqrt{3}}일 때 성립한다.

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