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*Fun* Polynomial Counting!

Source: 2023 AIME I #9

February 8, 2023
AMCAIMEAIME Ialgebrapolynomial

Problem Statement

Find the number of cubic polynomials p(x)=x3+ax2+bx+cp(x) = x^3 + ax^2 + bx + c, where aa, bb, and cc are integers in {−20,−19,−18,…,18,19,20}\{-20, -19,-18, \dots , 18, 19, 20\}, such that there is a unique integer m≠2m \neq 2 with p(m)=p(2)p(m) = p(2).
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