IMC 2005 · Problem 5

Find all $r > 0$ such that whenever $f : \mathbb{R}^2 \to \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)| = 1$ and $|\operatorname{grad} f(u) - \operatorname{grad} f(v)| \le |u - v|$ for all $u, v \in \mathbb{R}^2$, then the maximum of $f$ on the disk $\{u \in \mathbb{R}^2 : |u| \le r\}$ is attained at exactly one point.

($\operatorname{grad} f(u) = (\partial_1 f(u), \partial_2 f(u))$ is the gradient vector of $f$ at the point $u$. For a vector $u = (a,b)$, $|u| = \sqrt{a^2 + b^2}$.)