Question

Cho a,b > 0, ab > 2013a + 2014b

CMR:\(a+b>\left(\sqrt{2013}+\sqrt{2014}\right)^2\)

Answer 1

Hình như là CMR >\(A+B>\left(\sqrt{2013}+\sqrt{2014}\right)^2\)

Do \(ab>2013a+2014b\)

\(\Rightarrow1>\frac{2013}{b}+\frac{2014}{a}\)

\(\Rightarrow a+b>\frac{2013}{b}\left(a+b\right)+\frac{2014}{a}\left(a+b\right)=2013+\frac{2013a}{b}+\frac{2014b}{a}+2014\)

Áp dụng BĐT Cô si với a,b>0 ta có:

\(\frac{2013a}{b}+\frac{2014b}{a}\ge2\sqrt{\frac{2013a}{b}.\frac{2014b}{a}}=2\sqrt{2013.2014}\)

\(\Rightarrow a+b>2013+2\sqrt{2013.2014}+2014=\left(\sqrt{2013}+\sqrt{2014}\right)^2\)

Answer 2

(căn 2013+2014)² các bạn