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Consider 6 distinct points in a plane. Let $m$ and $M$ be the minimum and maximum distances between any pair of points. Show that $M/m \ge \sqrt3$.

I am more interested in arrangement of these points where the equality holds.

I figured out that the least possible value of $M/m$ should happen when the six points lie on corners and centre of a regular pentagon. But this gave me $2cos18 = \sqrt3.6$. How do I get $\sqrt3$?

Just a hint will suffice. Thanks in advance.

  • I am somehow feeling that my construction gives the least value of $M/m$. I am beginning to doubt the validity of the $\sqrt3$ bound (but the source is reliable). –  May 02 '13 at 08:24
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    http://www2.stetson.edu/~efriedma/maxmin/ suggests you have the correct construction. The bound follows, though it's not saturated. There's likely a simpler argument giving your bound. – not all wrong May 02 '13 at 08:33
  • Guess I was correct. Thanks. –  May 02 '13 at 08:41
  • Wenn you just write out function names like that, $\TeX$ interprets that as a juxtaposition of variable names and formats it accordingly. To get the appropriate font and spacing, you can use predefined commands like \cos, or, if you need an operator name for which there isn't a predefined command, you can use \operatorname{name}. Also note that you need to group expressions like $3.6$ using braces ({}) to use them as the parameter of a command like \sqrt. Also, presumably you mean $2\cos18^\circ$, not $2\cos18$, and this is $\approx\sqrt{3.6}$, not $=\sqrt{3.6}$. – joriki May 04 '13 at 16:14

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Community wiki answer so the question can be marked as answered: Apparently $2\cos\pi/10$ is indeed the minimal value of the ratio.

joriki
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