Question 92
Find all non-constant functions $f : \mathbb{R} \to \mathbb{R}$ such that:
(i) $f(x)f(x+y) = f(x)^2 + x \cdot f_{2025}(y)$ for all $x, y \in \mathbb{R}$,
(ii) $f_{2025}(\cos x) + f_{2025}(\cos(\pi - x)) = 0$ for all $x \in \mathbb{R}$,
where $f_n(x) = f(f(\ldots(f(x))\ldots))$ is the composition of the function $f$ with itself $n$ times.
(i) $f(x)f(x+y) = f(x)^2 + x \cdot f_{2025}(y)$ for all $x, y \in \mathbb{R}$,
(ii) $f_{2025}(\cos x) + f_{2025}(\cos(\pi - x)) = 0$ for all $x \in \mathbb{R}$,
where $f_n(x) = f(f(\ldots(f(x))\ldots))$ is the composition of the function $f$ with itself $n$ times.