Problem

Source: 2019 Belarusian National Olympiad 11.2

Tags: algebra, polynomial



The polynomial $$ Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2 $$is represented as the sum of squares of four polynomials of four variables with integer coefficients. a) Find at least one such representation b) Prove that for any such representation at least one of the four polynomials isidentically zero. (A. Yuran)