Problem

Source: First JBMO TST of France 2020, Problem 3

Tags: function, algebra



Let n be a nonzero natural number. We say about a function f ∶ R ⟶ R that is n-positive if, for any real numbers $x_1, x_2,...,x_n$ with the property that $x_1+x_2+...+x_n = 0$, the inequality $f(x_1)+f(x_2)+...+f(x_n)=>0$ is true a) Is it true that any 2020-positive function is also 1010-positive? b) Is it true that any 1010-positive function is 2020-positive?