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Find n such that there exist two polynomials f, g

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September 20, 2010
algebrapolynomialfunctional equationIMO Shortlist

Problem Statement

For which positive integers nn do there exist two polynomials ff and gg with integer coefficients of nn variables x1,x2,…,xnx_1, x_2, \ldots , x_n such that the following equality is satisfied:
∑i=1nxif(x1,x2,…,xn)=g(x12,x22,…,xn2) ?\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ?
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