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2021 Fall AMC 10A Problems/Problem 25

Revision as of 18:48, 22 November 2021 by Leo.euler (talk | contribs) (Created page with "== Problem == A quadratic polynomial with real coefficients and leading coefficient <math>1</math> is called <math>\emph{disrespectful}</math> if the equation <math>p(p(x))=0<...")
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Problem

A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?

$\textbf{(A) } \frac{5}{16} \qquad\textbf{(B) } \frac{1}{2} \qquad\textbf{(C) } \frac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \frac{9}{8}$

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