2021 AMC 12A Fall Problem 23

Question

A quadratic polynomial 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

\tilde{p}(x)

for which the sum of the roots is maximized. What is

\tilde{p}(1)?

\dfrac{5}{16}

\dfrac{1}{2}

\dfrac{5}{8}

1

\dfrac{9}{8}

If

p

has roots

r, s,

then

p(p(x)) = 0

means

p(x) = r

or

p(x) = s;

exactly three roots requires one of these to have a double root

With

(r - s)^2 + 4s = 0

and letting

u = r - s,

maximize

r + s = -\tfrac{u^2}{2} + u

Let

p

have roots

r

and

s.

Then

p(p(x)) = 0

splits into

p(x) = r

and

p(x) = s,

with discriminants

(r - s)^2 + 4r

and

(r - s)^2 + 4s.

Exactly three real roots means one discriminant is

0

and the other positive.

Take

(r - s)^2 + 4s = 0

and set

u = r - s.

Then

s = -\tfrac{u^2}{4}

and the other discriminant is

4u,

so

u>0.

Also

r+s=-\tfrac{u^2}{2}+u,

maximized at

u=1,

giving

r=\tfrac34

and

s=-\tfrac14.

So

\tilde{p}(x) = \left(x - \tfrac34\right)\left(x + \tfrac14\right),

and

\tilde{p}(1) = \tfrac14\cdot\tfrac54 = \tfrac{5}{16}.

Thus, the correct answer is A.

Problem 23 in other years