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x_{n+1}=f(x_n) where f(x) = 3(|x|+|x-1|-|x+1|)

Source: 2006 VMEO III Shortlist SL A8 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

November 24, 2021
algebra

Problem Statement

Let f(x)=3(∣x∣+∣x−1∣−∣x+1∣)f(x) = 3(|x|+|x-1|-|x+1|) and let xn+1=f(xn)x_{n+1}=f(x_n) ∀n≥0\forall n \ge 0. How many real number x0x_0 are there, that satisfy x0=x2007x_0=x_{2007} and x0,x1,x2,...,x2006x_0,x_1,x_2,...,x_{2006} are distinct?
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