Cho a,b,c là 3 số thực dương thỏa mãn : ab+bc+ac=1
Cm: $\frac{a}{\sqrt{a^{2}+1}} + \frac{b}{\sqrt{b^{2}+1 }} + \frac{c}{\sqrt{c^{2}+1 }}\leq \frac{3}{2}$
Dinh Xuan Hung Bạn chú ý cách đặt tiêu đề
$\frac{a}{\sqrt{a^2+1}}=\frac{a}{\sqrt{a^2+ab+bc+ac}}=\frac{a}{\sqrt{(a+b)(a+c)}}\leq \frac{a}{2}\left ( \frac{1}{a+b}+\frac{1}{a+c} \right )=\frac{1}{2}\left ( \frac{a}{a+b}+\frac{a}{a+c} \right )$
CMTT:$\frac{b}{\sqrt{b^2+1}}\leq \frac{1}{2}\left ( \frac{b}{a+b}+\frac{b}{b+c} \right )$
$\frac{c}{\sqrt{c^2+1}}\leq \frac{1}{2}\left ( \frac{c}{a+c}+\frac{c}{b+c} \right )$
$\Rightarrow \sum \frac{a}{\sqrt{a^2+1}}\leq \frac{1}{2}\left ( \sum \frac{a+b}{a+b} \right )=\frac{3}{2}$