Math Gold Medalist

Lor

2023 AIME I 

Problem 9

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ with $p(m) = p(2).$

Factorization

Quadratic Equation

p(m)-p(2)=0

Factorize p(m)-p(2)

If the roots of p(m)-p(2) is m1, m2 then

case1) m1=m2 (not equal to 2)

case2) m1 is not equal to m2=2

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   Solution