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2021 Algebra/NT #9: Cubic applied to itself solutions

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May 30, 2021
algebra

Problem Statement

Let ff be a monic cubic polynomial satisfying f(x)+f(āˆ’x)=0f(x) + f(-x) = 0 for all real numbers xx. For all real numbers yy, define g(y)g(y) to be the number of distinct real solutions xx to the equation f(f(x))=yf(f(x)) = y. Suppose that the set of possible values of g(y)g(y) over all real numbers yy is exactly {1,5,9}\{1, 5, 9\}. Compute the sum of all possible values of f(10)f(10).
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