IMC 2025 Problem 1

Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\operatorname{deg}(P) \geq 2$ . For every $x \in \mathbb{R}$ , let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point ( $x, P(x)$ ). a) Suppose that the degree of $P$ is odd. Show that $\bigcup_{x \in \mathbb{R}} \ell_x=\mathbb{R}^2$ . b) Does there exist a polynomial of even degree for which the above equality still holds?