ELMO 2024 Problem 3

For some positive integer $n,$ Elmo writes down the equation $$x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.$$ Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation $$f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.$$ Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$ . Is it possible that there is no integer $k$ (possibly depending on $f$ ) such that $f^k(x)=x$ for all $x$ ?

Srinivas Arun