Best Answer bushindo, 12 April 2013 - 02:43 AM

Spoiler for PROOF pt.3

I believe there is a subtle error here

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Guest Message by DevFuse

Started by BMAD, Apr 06 2013 03:30 AM

Best Answer bushindo, 12 April 2013 - 02:43 AM

Spoiler for PROOF pt.3

I believe there is a subtle error here

Spoiler for

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20 replies to this topic

Posted 06 April 2013 - 03:30 AM

The 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 . Assume the land is flat.

Posted 06 April 2013 - 12:01 PM

Can the arrangment be 3-D? i guess not.

Posted 07 April 2013 - 09:33 AM

Spoiler for Close

*Vidi vici veni.*

Posted 08 April 2013 - 05:47 AM

I would be interested to see the configuration that makes M = m * sqrt(3)

Spoiler for Graphical lower bound

*Vidi vici veni.*

Posted 09 April 2013 - 05:56 PM

Show that, unless the towns are all collinear, it is always possible to choose three of them so that they form a triangle with maximum angle of at least 120°.

Posted 09 April 2013 - 06:09 PM

I marked it solved originally but just realized you didn't show equality yet. Sorry.

I would be interested to see the configuration that makes M = m * sqrt(3)

Spoiler for Graphical lower bound

Posted 10 April 2013 - 07:21 PM

I think you should mark this as solved. There is no way to place 6 points on a surface in a way that the minimum distance between any 2 points was **m** and the maximum distance between any 2 points was **m*sqrt(3)**.

Spoiler for Informal proof that you can do at most 5

Posted 10 April 2013 - 07:47 PM

I am exploring one more possibility.

Spoiler for In my previous post

*Vidi vici veni.*

Posted 10 April 2013 - 08:58 PM

I am exploring one more possibility.

Spoiler for In my previous post

Spoiler for that radius is...

Posted 10 April 2013 - 09:48 PM

I won't spoiler this because it's not a solution but a refutation.

Consider a regular pentagon with side * t* and points at center and vertices.

The radius * r* of the circumcircle is the minimum distance among the six points.

The diagonal * d* is the maximum distance among the six points.

* t* = 2

* d* = PHI

M/n =* d*/

This is the answer from my post 3 where I conjectured a more dense arrangement of points.

I don't believe there is a more dense configuration of six points.

*Vidi vici veni.*

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