For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?
Source: Baltic Way 1998 Q15
Tags: inequalities, inequalities proposed
For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?