2023 IMO Shortlist A2

Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $$f(x+y) f(x-y) \geq f(x)^{2}-f(y)^{2}$$ for every $x, y \in \mathbb{R}$. Assume that the inequality is strict for some $x_{0}, y_{0} \in \mathbb{R}$. Prove that $f(x) \geq 0$ for every $x \in \mathbb{R}$ or $f(x) \leq 0$ for every $x \in \mathbb{R}$. (Malaysia)

令 $\mathbb{R}$ 为实数集。令 $f: \mathbb{R} \rightarrow \mathbb{R}$ 为一个函数，使得 $$f(x+y) f(x-y) \geq f(x)^{2}-f(y)^{2}$$ 对于每个 $x, y \in \mathbb{R}$。假设对于某些 $x_{0}, y_{0} \in \mathbb{R}$ 来说，不等式是严格的。证明 $f(x) \geq 0$ 对于每个 $x \in \mathbb{R}$ 或 $f(x) \leq 0$ 对于每个 $x \in \mathbb{R}$。 (马来西亚)