Library / Belarusian Mathematical Olympiad /111 of 208

Belarusian Mathematical Olympiad

Algebra Difficulty 6.1 National Olympiad Prove it Belarus

Is it possible to represent the polynomial of seven variables
$Q(x_1, x_2, \dots, x_7) = (x_1 + x_2 + \dots + x_7)^2 + 2(x_1^2 + x_2^2 + \dots + x_7^2)$
as a sum of squares of seven polynomials with non-negative integer coefficients:
$Q(x_1, \dots, x_7) = P_1(x_1, \dots, x_7)^2 + P_2(x_1, \dots, x_7)^2 + \dots + P_7(x_1, \dots, x_7)^2?$

Solution

Answer: yes, it is possible.
For example, consider the following representation:
$(x_1 + x_2 + \dots + x_7)^2 + 2(x_1^2 + x_2^2 + \dots + x_7^2) = (x_1 + x_2 + x_4)^2 + \\ \qquad +(x_2 + x_3 + x_5)^2 + (x_3 + x_4 + x_6)^2 + (x_4 + x_5 + x_7)^2 + \\ \qquad +(x_5 + x_6 + x_1)^2 + (x_6 + x_7 + x_2)^2 + (x_7 + x_1 + x_3)^2.$

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.