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# [SOLVED] Points on a disc

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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

Euclid

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 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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 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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 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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 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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 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.

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:32 am Poor Peyman! with his lots of spaces jut got a contradiction, don't trust your eyes Peyman"so you can surely find a disk of radius 1/3 that does not contain any of the six points"

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:35 am Mohammad wrote: Poor Peyman! with his lots of spaces jut got a contradiction, don't trust your eyes Peyman"so you can surely find a disk of radius 1/3 that does not contain any of the six points" Be nice or I'll ban you for 24 hours ...can you tell whether I'm kidding?Last edited by Bruno on Wed Nov 25, 2009 12:36 am; edited 1 time in total

Descartes

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 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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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:38 am

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 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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 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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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:46 am I'm looking forward to it .

Descartes

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:53 am More details:

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:55 am

Euclid

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:56 am what's up with the pictures?!

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 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!

Descartes

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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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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:04 am It's a great idea.

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 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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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:06 am peyman wrote:It's a great idea. Thanks. Congrats on the promotion to "Euclid".

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:07 am No promotion for me tonight

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 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

Descartes

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:09 am Ok, get ready to be bombarded

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 Subject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:10 am Mohammad wrote:Ok, get ready to be bombarded Get ready to be 24-hour banned. Just kidding, but take it to the Chit Chat! Spare my poor thread about six points on a disc!

 Subject: Re: [SOLVED] Points on a disc

 [SOLVED] Points on a disc
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