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Junior Member

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 Between six towns   « on: May 23rd, 2014, 3:08pm » Quote Modify

he smallest distance between any two of six towns is m miles. The largest distance between any two of the towns is M miles. Show that M/m > sqrt(3).
Assume the land is flat.
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Junior Member

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 Re: Between six towns   « Reply #1 on: May 31st, 2014, 9:09pm » Quote Modify

Bump
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dudiobugtron
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 Re: Between six towns   « Reply #2 on: Jun 1st, 2014, 4:07am » Quote Modify

One way to work this out would be to find the configuration which has the greatest ratio m:M, and show that even then it's not as great as 1:sqrt(3).

I couldn't initially figure out what that best configuration would be, though, so I gave up!  But since you bumped it, I'll think some more about it.
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pex
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 Re: Between six towns   « Reply #3 on: Jun 1st, 2014, 4:45am » Quote Modify

Apparently sqrt(3) is not even a tight bound. According to this page (spoiler alert: shows optimal configuration), the smallest possible value for M/m is sqrt( (5 + sqrt(5)) / 2 ) or approximately 1.90; sqrt(3) is approximately 1.73. Moreover, that page claims that the result is "trivial", but I have to admit I'm not seeing that.
 « Last Edit: Jun 1st, 2014, 4:47am by pex » IP Logged
Junior Member

Posts: 57
 Re: Between six towns   « Reply #4 on: Jun 1st, 2014, 5:57am » Quote Modify

Wouldn't the sqrt (3) be tighter since it is smaller?
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pex
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 Re: Between six towns   « Reply #5 on: Jun 1st, 2014, 6:31am » Quote Modify

on Jun 1st, 2014, 5:57am, BMAD wrote:
 Wouldn't the sqrt (3) be tighter since it is smaller?
No. That page claims that any configuration must have M/m > 1.9. If that's true, it immediately implies that any configuration also has M/m > sqrt(3), but not the other way around.

(By your reasoning, the bound M/m > 1 would be even tighter - but that's much easier to prove!)
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Junior Member

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 Re: Between six towns   « Reply #6 on: Jun 1st, 2014, 6:33am » Quote Modify

Oops. I got the inequality backwards in my head.
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Grimbal
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 Re: Between six towns   « Reply #7 on: Jun 1st, 2014, 2:38pm » Quote Modify

I think the optimal configuration is a pentagon with one city in the center.
The max distance is M=2*sin(2*pi/5)  assuming m=1.
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