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A5

Part of 2011 Putnam

Problems(1)

Putnam 2011 A5

Source:

12/5/2011
Let F:R2→RF:\mathbb{R}^2\to\mathbb{R} and g:R→Rg:\mathbb{R}\to\mathbb{R} be twice continuously differentiable functions with the following properties:
• F(u,u)=0F(u,u)=0 for every u∈R;u\in\mathbb{R};
• for every x∈R,g(x)>0x\in\mathbb{R},g(x)>0 and x2g(x)≤1;x^2g(x)\le 1;
• for every (u,v)∈R2,(u,v)\in\mathbb{R}^2, the vector ∇F(u,v)\nabla F(u,v) is either 0\mathbf{0} or parallel to the vector ⟨g(u),−g(v)⟩.\langle g(u),-g(v)\rangle.
Prove that there exists a constant CC such that for every n≥2n\ge 2 and any x1,…,xn+1∈R,x_1,\dots,x_{n+1}\in\mathbb{R}, we have min⁡i≠j∣F(xi,xj)∣≤Cn.\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.
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