ILL 1976 Problem 21

Find the largest positive real number $p$ (if it exists) such that the inequality $$x^2_1+ x_2^2+ \cdots + x^2_n\ge p(x_1x_2 + x_2x_3 + \cdots + x_{n-1}x_n)$$ is satisfied for all real numbers $x_i$ , and $(a) n = 2; (b) n = 5.$ Find the largest positive real number $p$ (if it exists) such that the inequality holds for all real numbers $x_i$ and all natural numbers $n, n \ge 2.$