Every problem from the world’s university-level science olympiads.

Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\deg(P) \ge 2$. 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 $\displaystyle\bigcup_{x \in \mathbb{R}} \ell_x = \mathbb{R}^2$.

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

Let $A$ and $B$ be real $n \times n$ matrices such that $A^2 + B^2 = AB$. Prove that if $BA - AB$ is an invertible matrix then $n$ is divisible by $3$.

Today, Ivan the Confessor prefers continuous functions $f \colon [0, 1] \to \mathbb{R}$ satisfying $f(x) + f(y) \ge |x - y|$ for all pairs $x, y \in [0, 1]$. Find the minimum of $\int_0^1 f$ over all preferred functions.