Question 82
The continuous function $f : \mathbb{R} \to \mathbb{R}$ has the property $(f \circ f)(x) = x^{2023}$, for every $x \in \mathbb{R}$. Prove that:
a) $\sqrt[2023]{f(x)} = f(\sqrt[2023]{x})$, for every $x \in \mathbb{R}$;
b) for every $n \ge 3$ there exists an arithmetic progression $x_1, x_2, \dots, x_n$ so that $\sum_{k=1}^{n} f(x_k) = 0$.
a) $\sqrt[2023]{f(x)} = f(\sqrt[2023]{x})$, for every $x \in \mathbb{R}$;
b) for every $n \ge 3$ there exists an arithmetic progression $x_1, x_2, \dots, x_n$ so that $\sum_{k=1}^{n} f(x_k) = 0$.