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Math Prize 2023 Problem 16

Source:

October 11, 2023

Problem Statement

Let f(x)=x2āˆ’3/4f(x) = x^2 - 3/4. Let f(n)(x)f^{(n)}(x) denote the composition of ff with itself nn times. For example, f(3)(x)=f(f(f(x)))f^{(3)}(x) = f(f(f(x))). Let RR be the set of complex numbers that is the union of the roots of the polynomials f(n)(x2+3/4)f^{(n)}(x^2 + 3/4) over positive integers nn. Let BB be the smallest rectangle in the complex plane with sides parallel to the real and imaginary axes that contains RR. What is the square of the area of BB?
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