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Omni-MATH / Let f be a monic cubic polynomial satisfying f(x)+f(−x)=0 for all real numbers x. For all real numbers y, define g(y) to be the number of distinct real solutions x to the equation …

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problem

Let ff be a monic cubic polynomial satisfying f(x)+f(−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).
Plain-text mathematical notation (without MathML)
Let f be a monic cubic polynomial satisfying f(x)+f(−x)=0 for all real numbers x. For all real numbers y, define g(y) to be the number of distinct real solutions x to the equation f(f(x))=y. Suppose that the set of possible values of g(y) over all real numbers y is exactly {1,5,9}. Compute the sum of all possible values of f(10).
Original LaTeX notation
Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1,5,9\}$. Compute the sum of all possible values of $f(10)$.

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Source and history

task_id
19c13379-32ee-57fa-8e3b-11b03df4794d
task_key
test--19c13379-32ee-57fa-8e3b-11b03df4794d
task_revision_id
1
config / split
/ test

Official source

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