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iterate function

Source: Brazil Undergrad MO 2021

March 17, 2021
functioncollege contestsquadraticsrootsFixed pointBrazilian Undergrad MOBrazilian Undergrad MO 2020

Problem Statement

Let f(x)=2x2+x−1,f0(x)=xf(x) = 2x^2 + x - 1, f^{0}(x) = x, and fn+1(x)=f(fn(x))f^{n+1}(x) = f(f^{n}(x)) for all real x>0x>0 and n≥0n \ge 0 integer (that is, fnf^{n} is ff iterated nn times).
a) Find the number of distinct real roots of the equation f3(x)=xf^{3}(x) = x b) Find, for each n≥0n \ge 0 integer, the number of distinct real solutions of the equation fn(x)=0f^{n}(x) = 0
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