Question

[Long Trần Bảo]

tìm 3 số thực dương x;y;z thỏa mãn \(\dfrac{2}{\sqrt{x}+2\sqrt{y}+3\sqrt{z}}-\dfrac{1}{2\sqrt{xy}+6\sqrt{yz}+3\sqrt{zx}}=\dfrac{1}{3}\)

Answer 1

[Việt Bắc Nguyễn]

Đặt

\(\left\{{}\begin{matrix}\sqrt{x}=a\\2\sqrt{y}=b\\3\sqrt{z}=c\end{matrix}\right.\)

\(\Rightarrow\frac{2}{a+b+c}-\frac{1}{ab+bc+ca}=\frac{1}{3}\)

\(\left(\sum a,\sum ab\right)\rightarrow\left(p,q\right)\)

Ta chứng minh :

\(\frac{2}{p}-\frac{1}{q}\le\frac{1}{3}\)

\(\Leftrightarrow p\ge\frac{6q}{q+3}\Leftrightarrow p^2\ge\frac{36q^2}{\left(q+3\right)^2}\)

Thấy : \(p^2\ge3q\)

Ta chứng minh :

\(3q\ge\frac{36q^2}{\left(q+3\right)^2}\Leftrightarrow\left(q-3\right)^2\ge0\)(luôn đúng).

\(\Rightarrow\)Dấu "=" xảy ra \(\Rightarrow a=b=c=1\)

\(\Rightarrow\left(x,y,z\right)\rightarrow\left(..,..,..\right)\)

#Kaito#