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Print69th Belarusian Mathematical Olympiad
Belarus algebra
Problem
Is it possible to represent the polynomial of seven variables as a sum of squares of seven polynomials with non-negative integer coefficients:
Solution
Answer: yes, it is possible. For example, consider the following representation:
Final answer
Yes. For example: (x1 + x2 + x3 + x4 + x5 + x6 + x7)^2 + 2(x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 + x7^2) = (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.
Techniques
Polynomial operationsSymmetric functions