MathDB
Disrespectful Polynomials

Source: 2021 November AMC 10A #25/ 12A #23

November 11, 2021
AMC 10algebrapolynomial

Problem Statement

A quadratic polynomial p(x)p(x) with real coefficients and leading coefficient 11 is called disrespectful if the equation p(p(x))=0p(p(x)) = 0 is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial p~(x)\tilde{p}(x) for which the sum of the roots is maximized. What is p~(1)?\tilde{p}(1)?
<spanclass=latexbold>(A)</span>516<spanclass=latexbold>(B)</span>12<spanclass=latexbold>(C)</span>58<spanclass=latexbold>(D)</span>1<spanclass=latexbold>(E)</span>98<span class='latex-bold'>(A) </span>\dfrac5{16} \qquad <span class='latex-bold'>(B) </span>\dfrac12 \qquad <span class='latex-bold'>(C) </span>\dfrac58 \qquad <span class='latex-bold'>(D) </span>1 \qquad <span class='latex-bold'>(E) </span>\dfrac98