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Problem 3 of Third round

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

September 9, 2022
algebra

Problem Statement

The positive integers pp, qq are such that for each real number xx (x+1)p(x−3)q=xn+a1xn−1+a2xn−2+⋯+an−1x+an(x+1)^p (x-3)^q=x^n+a_1 x^{n-1}+a_2 x^{n-2}+\dots +a_{n-1} x+a_n where n=p+qn=p+q and a1,…,ana_1,\dots ,a_n are real numbers. Prove that there exists infinitely many pairs (p,q)(p,q) for which a1=a2a_1=a_2.
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