1 reply to this topic

[mrjackass]

Cho $a,b,c \geq 0$. CMR:
$\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a} \leq\frac{3(a^2+b^2+c^2)}{a+b+c}$

[sieutoan99]

Cho $a,b,c \geq 0$. CMR:
$\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a} \leq\frac{3(a^2+b^2+c^2)}{a+b+c}$

Ta có:
BDT$\Leftrightarrow (a+b+c)(\frac{a^2+b^2}{a+b}+\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a})\leq 3(a^2+b^2+c^2)$
$\Leftrightarrow \frac{c(a^2+b^2)}{a+b}+\frac{a(b^2+c^2)}{b+c}+\frac{b(c^2+a^2)}{c+a}\leq a^2+b^2+c^2$
$\Leftrightarrow c^2-\frac{c(a^2+b^2)}{a+b}+a^2-\frac{a(b^2+c^2)}{b+c}+b^2-\frac{b(c^2+a^2)}{c+a}\geq 0$
$\Leftrightarrow \frac{ac(c-a)}{a+b}+\frac{bc(c-b)}{a+b}+\frac{ab(a-b)}{b+c}+\frac{ac(a-c)}{b+c}+\frac{ab(b-a)}{c+a}+\frac{bc(b-c)}{c+a}\geq 0$
$\Leftrightarrow \frac{ac(c-a)^2}{(a+b)(b+c)}+\frac{bc(c-b)^2}{(a+b)(a+c)}+\frac{ab(b-a)^2}{(c+a)(b+c)}\geq 0$ (BDT này luôn đúng)
Dấu "=" xảy ra khi a=b=c

Edited by sieutoan99, 07-02-2013 - 15:19.