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\(Q=\sum\dfrac{\left(a+b\right)^2}{\sqrt{2\left(b+c\right)^2+bc}}\ge\sum\dfrac{\left(a+b\right)^2}{\sqrt{2\left(b+c\right)^2+\dfrac{1}{4}\left(b+c\right)^2}}=\dfrac{2}{3}\sum\dfrac{\left(a+b\right)^2}{b+c}\)
\(Q\ge\dfrac{2}{3}.\dfrac{\left(a+b+b+c+c+a\right)^2}{a+b+b+c+c+a}=\dfrac{4}{3}\left(a+b+c\right)=\dfrac{4}{3}\)
ques này nhiều ng` hỏi r` thay ab+bc+ca=1 vào rồi phân tích rút gọn
Do ab + bc + ca = 1 nên ta có :
\(a\sqrt{\frac{\left(b^2+1\right)\left(c^2+1\right)}{a^2+1}}=a\sqrt{\frac{\left(b^2+ab+ac+bc\right)\left(c^2+ab+ac+bc\right)}{a^2+ab+ac+bc}}\)
\(=a\sqrt{\frac{\left(a+b\right)\left(b+c\right)\left(a+c\right)\left(b+c\right)}{\left(a+b\right)\left(a+c\right)}}=a\sqrt{\left(b+c\right)^2}=a\left(b+c\right)=ab+ac\text{ }\left(1\right)\)
Tương tự : \(b\sqrt{\frac{\left(a^2+1\right)\left(c^2+1\right)}{b^2+1}}=ab+bc\) (2)và \(c\sqrt{\frac{\left(b^2+1\right)\left(a^2+1\right)}{c^2+1}}=bc+ac\) (3)
Cộng vế với vế của (1) ; (2) ; (3) lại ta được :
\(a\sqrt{\frac{\left(b^2+1\right)\left(c^2+1\right)}{a^2+1}}+b\sqrt{\frac{\left(a^2+1\right)\left(c^2+1\right)}{b^2+1}}+c\sqrt{\frac{\left(b^2+1\right)\left(a^2+1\right)}{c^2+1}}=2\left(ab+bc+ac\right)=2\)
Ta có : \(b=\dfrac{c+a}{2}\Rightarrow2b=c+a\Rightarrow a-b=b-c\)
Dó đó : \(P=\left(\dfrac{1}{\sqrt{a}+\sqrt{b}}+\dfrac{1}{\sqrt{b}+\sqrt{c}}\right)\left(\sqrt{a}+\sqrt{c}\right)\)
\(P=\left[\dfrac{\sqrt{a}-\sqrt{b}}{\left(\sqrt{a}+\sqrt{b}\right)\left(\sqrt{a}-\sqrt{b}\right)}+\dfrac{\sqrt{b}-\sqrt{c}}{\left(\sqrt{b}+\sqrt{c}\right)\left(\sqrt{b}-\sqrt{c}\right)}\right]\left(\sqrt{a}+\sqrt{c}\right)\)
\(P=\left[\dfrac{\sqrt{a}-\sqrt{b}}{a-b}+\dfrac{\sqrt{b}-\sqrt{c}}{b-c}\right]\left(\sqrt{a}+\sqrt{c}\right)\)
\(P=\left[\dfrac{\sqrt{a}-\sqrt{b}}{b-c}+\dfrac{\sqrt{b}-\sqrt{c}}{b-c}\right]\left(\sqrt{a}+\sqrt{c}\right)\) Vì \(\left(a-b=b-c\right)\)
\(P=\left[\dfrac{\sqrt{a}-\sqrt{b}+\sqrt{b}-\sqrt{c}}{b-c}\right]\left(\sqrt{a}+\sqrt{c}\right)\)
\(P=\dfrac{\sqrt{a}-\sqrt{c}}{b-c}\left(\sqrt{a}+\sqrt{c}\right)\)
\(P=\dfrac{a-c}{a-b}=\dfrac{a-c}{a-\dfrac{a+c}{2}}=\dfrac{a-c}{\dfrac{2a-a-c}{2}}=\dfrac{a-c}{\dfrac{a-c}{2}}=2\)