Áp dụng bất đẳng thức Cauchy cho 3 bộ số thực không âm:
\(\Rightarrow\left\{\begin{matrix}3yzt\le y^3+z^3+t^3\\3xtz\le x^3+t^3+z^3\\3xyt\le x^3+y^3+t^3\\3xyz\le x^3+y^3+z^3\end{matrix}\right.\)
\(\Rightarrow\left\{\begin{matrix}x^3+3yzt\le x^3+y^3+z^3+t^3\\y^3+3xtz\le x^3+y^3+z^3+t^3\\z^3+3xyt\le x^3+y^3+z^3+t^3\\t^3+3xyz\le x^3+y^3+z^3+t^3\end{matrix}\right.\)
\(\Rightarrow\left\{\begin{matrix}\frac{x^3}{x^3+3yzt}\ge\frac{x^3}{x^3+y^3+z^3+t^3}\\\frac{y^3}{y^3+3xtz}\ge\frac{y^3}{x^3+y^3+z^3+t^3}\\\frac{z^3}{z^3+3xyt}\ge\frac{z^3}{x^3+y^3+z^3+t^3}\\\frac{t^3}{t^3+3xyz}\ge\frac{t^3}{x^3+y^3+z^3+t^3}\end{matrix}\right.\)
\(\Rightarrow\frac{x^3}{x^3+3yzt}+\frac{y^3}{y^3+3xtz}+\frac{z^3}{z^3+3xyt}+\frac{t^3}{t^3+3xyz}\ge\frac{x^3+y^3+z^3+t^3}{x^3+y^3+z^3+t^3}\)
\(\Rightarrow\frac{x^3}{x^3+3yzt}+\frac{y^3}{y^3+3xtz}+\frac{z^3}{z^3+3xyt}+\frac{t^3}{t^3+3xyz}\ge1\) ( đpcm )
