Cho a, b, c > 0. CM:
a) $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\geq a+b+c$
b) $\frac{a^2}{b^2+c^2}+\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}\geq\frac{a}{b+c}+\frac{b}{c+a} +\frac{c}{a+b}$
Cho a, b, c > 0. CM:
a) $\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\geq a+b+c$
b) $\frac{a^2}{b^2+c^2}+\frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}\geq\frac{a}{b+c}+\frac{b}{c+a} +\frac{c}{a+b}$
Giải thích các bước giải:
Áp dụng BĐT cosi với \(a,b,c>0\), ta có:
\( \dfrac{bc}a+\dfrac{ca}b\ge 2\sqrt{\dfrac{bc}a\cdot \dfrac{ca}{b}}=2\sqrt{c^2}=2c\)
CMTT: \(\begin{cases}\dfrac{ab}c+\dfrac{ca}{b}\ge 2a\\ \dfrac{bc}a+\dfrac{ca}{b}\ge 2c\end{cases}\)
\(\to \dfrac{bc}a+\dfrac{ca}{b}+\dfrac{ab}c+\dfrac{ab}c+\dfrac{bc}a+\dfrac{ca}b\ge 2a+2b+2c\\\to \dfrac{bc}a+\dfrac{ca}b+\dfrac{ab}c\ge a+b+c\)