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Algebra Difficulty 6.1 National olympiad Prove it Belarus

Is it possible to represent the polynomial of seven variables
Q(x1,x2,…,x7)=(x1+x2+⋯+x7)2+2(x12+x22+⋯+x72) 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(x1,…,x7)=P1(x1,…,x7)2+P2(x1,…,x7)2+⋯+P7(x1,…,x7)2? 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:
(x1+x2+⋯+x7)2+2(x12+x22+⋯+x72)=(x1+x2+x4)2++(x2+x3+x5)2+(x3+x4+x6)2+(x4+x5+x7)2++(x5+x6+x1)2+(x6+x7+x2)2+(x7+x1+x3)2. (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.

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