Problem of the week

2021-16 Optimal constant

For a given positive integer $$n$$ and a real number $$a$$, find the maximum constant $$b$$ such that
$x_1^n + x_2^n + \dots + x_n^n + a x_1 x_2 \dots x_n \geq b (x_1 + x_2 + \dots + x_n)^n$
for any non-negative $$x_1, x_2, \dots, x_n$$.