International Mathematics Competition
for University Students
2025

IMC2025: Day 1, Problem 1

Problem 1. Let \(\displaystyle P\in\mathbb{R}[x]\) be a polynomial with real coefficients, and suppose \(\displaystyle \operatorname{deg}(P)\geq 2\). For every \(\displaystyle x \in \mathbb{R}\), let \(\displaystyle \ell_x \subset \mathbb{R}^2\) denote the line tangent to the graph of \(\displaystyle P\) at the point \(\displaystyle ( x, P(x) )\).

(a) Suppose that the degree of \(\displaystyle P\) is odd. Show that \(\displaystyle \displaystyle\bigcup_{x \in \mathbb{R}} \ell_x = \mathbb{R}^2.\)

(b) Does there exist a polynomial of even degree for which the above equality still holds?

Mike Daas, Max Planck Institute for Mathematics, Bonn

  Solution