| | [SOLVED] Points on a disc |  |
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Bruno
Admin
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| | Subject: [SOLVED] Points on a disc Tue Nov 24, 2009 6:05 pm | |
| This problem is my own. It's quite easy. Hope you like it!
Suppose you are given 6 points on a disc of radius 1. Show that inside this disc you can fit a smaller disc of radius 1/3 containing none of the points.
Last edited by Bruno on Wed Nov 25, 2009 12:52 am; edited 1 time in total | |
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peyman
Euclid
Posts : 49
Join date : 2009-11-06
Age : 100
Location : dense in the universe
| | Subject: Re: [SOLVED] Points on a disc Tue Nov 24, 2009 11:10 pm | |
| This is how I would go about solving it, but I don't really want to work out the details:
Suppose the unit circle is centered at the origin.
Suppose you canNOT fit a disk of radius 1/3 in the bigger disk, then every disk of radius 1/3 must contain one of the 6 points.
Consider 5 circles with radius 1/3 centered at (2/3,0),(0,2/3),(-2/3,0),(0,-2/3) and (0,0).
Each must contain at least one point.
But then you have only one more point and lot's of space to cover so you can surely find a disk of radius 1/3 that does not contain any of the six points.
This is not a proof of course! | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:11 am | |
| Peyman: Use this a lot of space to give a lot of proofs  Problem is not correct check your proof Bruno! | |
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Bruno
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:15 am | |
| It is correct! I'll bet the drink you owe me that it is.  | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:20 am | |
| Take 5 points uniformly distributed on circle with radius 2/3 and with center at (0,0) and the sixth point out of this circle, there is no way to put them all in one circle with radius 1/3 even 2/3 . I know you can find 7 circles in the main circle but they kiss each other and they are not disjoint
Last edited by Mohammad on Wed Nov 25, 2009 12:24 am; edited 1 time in total | |
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Bruno
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Posts : 184
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:23 am | |
| - Mohammad wrote:
- I know you can find 7 circles in the main circle but they kiss each other and they are not disjoint:!:
Oh then you've got it! I should have been more precise : you can find a disc of radius 1/3 whose inside contains none of the points. ![[SOLVED] Points on a disc 2d](http://local.wasp.uwa.edu.au/~pbourke/geometry/kissing/2d.gif) | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:32 am | |
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Bruno
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:35 am | |
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Last edited by Bruno on Wed Nov 25, 2009 12:36 am; edited 1 time in total | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:36 am | |
| Oh, I forgot he is wise old man  | |
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Bruno
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Posts : 184
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:38 am | |
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peyman
Euclid
Posts : 49
Join date : 2009-11-06
Age : 100
Location : dense in the universe
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:43 am | |
| I don't understand your reasoning mohammad. | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:44 am | |
| Don't mind, I'll explain to you tomorrow  Nice pic dude | |
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peyman
Euclid
Posts : 49
Join date : 2009-11-06
Age : 100
Location : dense in the universe
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:46 am | |
| I'm looking forward to it  . | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:53 am | |
| More details: ![[SOLVED] Points on a disc Img_1011](https://i.servimg.com/u/f87/14/57/86/12/th/img_1011.jpg) | |
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Bruno
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Posts : 184
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Location : the infinite, frictionless plane of uniform density
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:55 am | |
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peyman
Euclid
Posts : 49
Join date : 2009-11-06
Age : 100
Location : dense in the universe
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 12:56 am | |
| what's up with the pictures?! | |
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Bruno
Admin
Posts : 184
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Location : the infinite, frictionless plane of uniform density
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:00 am | |
| I don't know what's going on. I think Mo figured it would help him convince you that he is right if he posted a picture of him looking at you straight in the eyes. I think we need a thread for nonsense only somewhere. It's good to let it out!  | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
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| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:02 am | |
| Hey Bruno, you got the point !Looking straight to Peyman, I just took the picture very hot! | |
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peyman
Euclid
Posts : 49
Join date : 2009-11-06
Age : 100
Location : dense in the universe
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:04 am | |
| It's a great idea.  | |
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Bruno
Admin
Posts : 184
Join date : 2009-09-15
Age : 36
Location : the infinite, frictionless plane of uniform density
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:06 am | |
| - Mohammad wrote:
- Hey Bruno, you got the point !Looking straight to Peyman, I just took the picture very hot!
Yeah I'm getting used to your short proofs.  | |
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Bruno
Admin
Posts : 184
Join date : 2009-09-15
Age : 36
Location : the infinite, frictionless plane of uniform density
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:07 am | |
| No promotion for me tonight  | |
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Bruno
Admin
Posts : 184
Join date : 2009-09-15
Age : 36
Location : the infinite, frictionless plane of uniform density
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:08 am | |
| - Mohammad wrote:
- No promotion for me tonight
You always want to be one step ahead now don't you. I think the next promotion is at 100 posts or something close.  Of course, I don't need a promotion because I'm already the forum god.
Last edited by Bruno on Wed Nov 25, 2009 1:09 am; edited 1 time in total | |
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Mohammad
Descartes
Posts : 100
Join date : 2009-11-05
Age : 40
Location : Right behind you
| | Subject: Re: [SOLVED] Points on a disc Wed Nov 25, 2009 1:09 am | |
| Ok, get ready to be bombarded | |
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Bruno
Admin
Posts : 184
Join date : 2009-09-15
Age : 36
Location : the infinite, frictionless plane of uniform density
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