Putnam 2011 A5

Let $F : \mathbb{R}^2 \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be twice continuously differentiable functions with the following properties:

• $F(u,u) = 0$ for every $u \in \mathbb{R}$;

• for every $x \in \mathbb{R}$, $g(x) > 0$ and $x^2 g(x) \leq 1$;

• for every $(u,v) \in \mathbb{R}^2$, the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u), -g(v) \rangle$.

Prove that there exists a constant $C$ such that for every $n\geq 2$ and any $x_1,\dots,x_{n+1} \in \mathbb{R}$, we have $$\min_{i \neq j} |F(x_i,x_j)| \leq \frac{C}{n}.$$