Problem:
A quadratic polynomial p(x) with real coefficients and leading coefficient 1 is called 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 p~​(x) for which the sum of the roots is maximized. What is p~​(1) ?
Answer Choices:
A. 165​
B. 21​
C. 85​
D. 1
E. $$\frac{9}{8}$$
Solution:
Suppose p(x)=(x−r)(x−s). Observe that p(x) must have (two) real roots in order for p(p(x)) to have any roots at all. More specifically, if y is a root of p(p(x)), then p(y)=r or p(y)=s. That is, the equations
(x−r)(x−s)−r=0 and (x−r)(x−s)−s=0
together must have exactly three real roots among them. It follows that one of these two quadratics, say (x−r)(x−s)−r, must have discriminant zero.\
Expansion yields x2−(r+s)x+r(s−1)=0, so the discriminant Δ of this quadratic must satisfy
0=Δ=(r+s)2−4r(s−1)=(r−s)2+4r.
This implies that r is negative, say r=−r0​, and that s=r±−4r​=−r0​±2r0​​. It follows that
where the second inequality follows from the fact that a−a2≤41​ for all real numbers a. Thus r=−41​ and s=43​, which works. In turn, p~​(x)=(x+41​)(x−43​) and p~​(1)=45​⋅41​=(A)165​​.