BG-MO 1974 Problem 3

(a) Find all real numbers $p$ for which the inequality $$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$ is true for all real numbers $x_1,x_2,x_3$ . (b) Find all real numbers $q$ for which the inequality $$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$ is true for all real numbers $x_1,x_2,x_3,x_4$ .

I. Tonov