Problem

Source: 2015 Brazil IMO TST 3.1

Tags: function, odd, Even, periodic, algebra



Let's call a function f:RR 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.