[TEX]x_1,x_2.x_3,x_4 ......x_n > 0[/TEX] chứng minh rằng
[TEX]\frac{1}{x_1^n+x_2^n+x_3^n+x_4^n+...+ x_{n-1}^n+x_1x_2x_3x_4..x_n}+ \frac{1}{x_1^n+x_2^n+x_3^n+x_4^n+...+ x_{n-2}^n+x_n^n+x_1x_2x_3x_4..x_n}+\frac{1}{x_1^n+x_2^n+x_3^n+x_4^n+...+ x_{n-1}^n+x_n^n+x_1x_2x_3x_4..x_n}+ \ldots+\frac{1}{x_2^n+x_3^n+x_4^n+...+x_{n-2}^n+ x_{n-1}^n+x_1x_2x_3x_4..x_n } \le \frac{1}{x_1x_2x_3x_4..x_n} [/TEX]
Chú ý rằng :
[TEX]x_1^n+x_2^n+x_3^n+x_4^n+...+ x_{n-1}^n+x_1x_2x_3x_4..x_n \ge x_1x_2x_3x_4..x_{n-1}\( x_1+x_2+x_3+ \ldots x_{n-1}\)+ x_1x_2x_3x_4..x_n= x_1x_2x_3x_4..x_{n-1}\(x_1+x_2+x_3+ \ldots x_{n-1}+x_n\)[/TEX]
Tương tự cọng lại thì ta có :
[TEX]VT \le \frac{1}{x_1x_2x_3x_4..x_{n-1}\(x_1+x_2+x_3+ \ldots x_{n-1}+x_n\)}+\frac{1}{x_1x_2x_3x_4..x_{n-2}x_n^n\(x_1+x_2+x_3+ \ldots x_{n-1}+x_n\)}+\frac{1}{x_1x_2x_3x_4..x_{n-3}^nx_{n-1}^nx_n^n\(x_1+x_2+x_3+ \ldots x_{n-1}+x_n\)} + \ldots \frac{1}{x_2x_3x_4..x_{n-1}x_n^n\(x_1+x_2+x_3+ \ldots x_{n-1}+x_n\)} =VP[/TEX]