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2016-2017 Fall OMO Problem 19

Source:

November 16, 2016
Online Math Open

Problem Statement

Let SS be the set of all polynomials Q(x,y,z)Q(x,y,z) with coefficients in {0,1}\{0,1\} such that there exists a homogeneous polynomial P(x,y,z)P(x,y,z) of degree 20162016 with integer coefficients and a polynomial R(x,y,z)R(x,y,z) with integer coefficients so that P(x,y,z)Q(x,y,z)=P(yz,zx,xy)+2R(x,y,z)P(x,y,z) Q(x,y,z) = P(yz,zx,xy)+2R(x,y,z) and P(1,1,1)P(1,1,1) is odd. Determine the size of SS.
Note: A homogeneous polynomial of degree dd consists solely of terms of degree dd.
Proposed by Vincent Huang
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