Problem

Source: 2015 Brazil IMO TST 3.1

Tags: function, odd, Even, periodic, algebra



Let's call a function $f : R \to R$ cool if there are real numbers $a$ and $b$ such that $f(x + a)$ is an even function and $f(x + b)$ is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.