Question

cho x,y,z >0 thỏa: x+y+z=1

cm:

\(\frac{350}{xy+yz+zx}+\frac{386}{x^2+y^2+z^2}>2015\)

Answer 1

Áp dụng BĐT Cauchy-Schwarz dạng Engelta có:

\(VT=\frac{700}{2\left(xy+yz+xz\right)}+\frac{386}{x^2+y^2+z^2}\)\(=\frac{\sqrt{700}^2}{2\left(xy+yz+xz\right)}+\frac{\sqrt{386}^2}{x^2+y^2+z^2}\)

\(\ge\frac{\left(\sqrt{700}+\sqrt{386}\right)^2}{x^2+y^2+z^2+2\left(xy+yz+xz\right)}\)\(=\frac{\left(\sqrt{700}+\sqrt{386}\right)^2}{\left(x+y+z\right)^2}\)

\(=\left(\sqrt{700}+\sqrt{386}\right)^2>2015\left(x+y+z=1\right)\)