https://olm.vn/thanhvien/chibiverycute là con chó

https://olm.vn/thanhvien/chibiverycute là con chó

\(L.H.S=\Sigma_{cyc}\frac{a^2}{b}=\Sigma_{cyc}\left(\frac{a^2}{b}-a+b\right)=\Sigma_{cyc}\frac{a^2-ab+b^2}{b}\)

\(=\Sigma_{cyc}\left(\frac{a^2-ab+b^2}{b}+b\right)-\left(a+b+c\right)\)

\(\ge2\Sigma_{cyc}\sqrt{a^2-ab+b^2}-\left(a+b+c\right)\)

\(=\Sigma_{cyc}\sqrt{a^2-ab+b^2}+\Sigma_{cyc}\sqrt{\frac{1}{4}\left(a+b\right)^2+\frac{3}{4}\left(a-b\right)^2}-\left(a+b+c\right)\)

\(\ge\Sigma_{cyc}\sqrt{a^2-ab+b^2}+\Sigma_{cyc}\sqrt{\frac{1}{4}\left(a+b\right)^2}-\left(a+b+c\right)=\Sigma_{cyc}\sqrt{a^2-ab+b^2}=R.H.S\)

Đẳng thức xảy ra khi a = b = c

HPT

\(\Leftrightarrow\hept{\begin{cases}\left(x^2+1\right)+y\left(x+y-2\right)=2y\\\left(x^2+1\right)\left(x+y-2\right)=y\end{cases}}\)

y=0 khong phai nghiem cua hpt

\(\Rightarrow\hept{\begin{cases}\left(\frac{x^2}{y}+\frac{1}{y}\right)+\left(x+y-2\right)=2\\\left(\frac{x^2}{y}+\frac{1}{y}\right)\left(x+y-2\right)=1\end{cases}}\)

Dat \(\hept{\begin{cases}\frac{x^2}{y}+\frac{1}{y}=a\\x+y-2=b\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}a+b=2\\ab=1\end{cases}}\)

Đến đây là ngon

\(x=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)

\(\Leftrightarrow x^3=\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)^3\)

\(\Leftrightarrow x^3=3+2\sqrt{2}+3-2\sqrt{2}+3\sqrt[3]{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}\left[\left(3+2\sqrt{2}\right)+\left(3-2\sqrt{2}\right)\right]\)

\(\Leftrightarrow x^3=6+3\sqrt[2]{9-8}.x\)

\(\Leftrightarrow x^3=6+3x\)