Let $P_{0},P_{1},P_{2},\cdots,P_{n}$ be $n+1$ points in the plane. Let $ d=1$ denote the minimal value of all the distances between any two points. Prove that


This problem background is from China high school math competition (Oct 14, 2012) problem 15,can see http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2822547&sid=bbbc81f99da00d657f61b4835931c87e#p2822547

also can see this two solution:http://wenku.baidu.com/view/82fb84d4240c844769eaeea3.html

But for my problem,I can't prove it.and I think this is nice problem,and Thank you for your help.


1 Answer 1


Up to relabelling, we can assume that $P_0 P_n$ is the greatest distance among $P_0 P_i$. We can also assume that $$\sum_{j=1}^{n-1}\frac{1}{P_0 P_j}<\sqrt{15(n-1)}$$ holds as induction hypothesis.

If $P_0 P_n \geq \frac{\sqrt{n}+\sqrt{n+1}}{\sqrt{15}}$ there is nothing to prove.

Otherwise, all the points belong to a circle $\Gamma$ centered in $P_0$ having radius $$R= \frac{\sqrt{n}+\sqrt{n+1}}{\sqrt{15}}<2\sqrt{\frac{n+1}{15}}$$ and area $[\Gamma]=\pi R^2$ less than $\frac{4\pi}{15}(n+1)$. Since at most $2\pi R$ points can lie in the annulus $A=\{x:d(x,P_0)\in[R-1/2,R]\}$, we have at least $(n+1-2\pi R)$ disjoint circles, having radius $\frac{1}{2}$, contained in $\Gamma$. By considering the area of the associated triangulation, we have: $$[\Gamma]= \pi R^2 > (n+1-2\pi R)\frac{\sqrt{3}}{2},$$ or: $$ \frac{2\pi}{\sqrt{3}}R^2 + 2\pi R > (n+1) $$ that is a contradiction once ... ahem ... $n>9881$. This leaves "a few" cases to be checked by hand.

  • 2
    $\begingroup$ Hi, Jack D'Aurizio. I cannot follow the step in the sentence beginning with "By considering the area of the associated triangulation". More precisely, I don't know how to construct an associated triangulation so that the coefficient $\frac{\sqrt{3}}{2}$ occurs on the right hand side of the inequality. Just in case you are interested, I posted a related question. $\endgroup$ Oct 25, 2013 at 14:00

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .