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Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that$$P(Q(x))=3Q(P(x))+1$$for all real numbers $x$.

Attempt:

I found a different construction $P(x) = \frac{3}{2}[(2x + 1)^{n+1}-1], Q(x) = \frac{3}{2}[(2x + 1)^n-1]$, which came from $3x[(2x+1)^n + (2x+1)^{n-1} + ... + 1]$. I guessed this construction from the small cases I did, and I feel that if (P, Q) works then (something, P) probably works, so I tried to inductively construct a pair of polynomials.

Any more complete and elegant solution?

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  • $\begingroup$ shouldn't "with degree no less than 2018" mean that all the degrees are $\ge 2018$? This means that increasing the degree doesn't lead to a contradiction $\endgroup$ Jan 7 '20 at 11:57
  • $\begingroup$ @CalvinKhor I added my recent attempt $\endgroup$
    – Behemooth
    Jan 7 '20 at 12:03

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