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 [SOLVED] Points on a disc

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Bruno
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PostSubject: [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
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PostSubject: 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
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PostSubject: 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 Exclamation
Problem is not correct Exclamation check your proof Bruno!
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PostSubject: 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. Wink
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PostSubject: 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 Exclamation. I know you can find 7 circles in the main circle but they kiss each other and they are not disjoint Exclamation


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PostSubject: 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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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:32 am

bounce 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" Exclamation bounce Exclamation
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:35 am

Mohammad wrote:
bounce 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" Exclamation bounce Exclamation

Be nice or I'll ban you for 24 hours jocolor


...

can you tell whether I'm kidding?


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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:36 am

Oh, I forgot he is wise old man Embarassed
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:38 am

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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:43 am

I don't understand your reasoning mohammad.
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:44 am

Don't mind, I'll explain to you tomorrow bounce Nice pic dude Exclamation
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:46 am

I'm looking forward to it Smile.
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:53 am

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

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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 12:56 am

what's up with the pictures?!
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PostSubject: 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! farao
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PostSubject: 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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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:04 am

It's a great idea. Very Happy
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PostSubject: 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. Razz
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:06 am

peyman wrote:
It's a great idea. Very Happy

Thanks. Congrats on the promotion to "Euclid". santa
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:07 am

No promotion for me tonight Question
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:08 am

Mohammad wrote:
No promotion for me tonight Question

You always want to be one step ahead now don't you.

I think the next promotion is at 100 posts or something close. afro Of course, I don't need a promotion because I'm already the forum god.


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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:09 am

Ok, get ready to be bombarded bounce
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PostSubject: Re: [SOLVED] Points on a disc   Wed Nov 25, 2009 1:10 am

Mohammad wrote:
Ok, get ready to be bombarded bounce

Get ready to be 24-hour banned. jocolor

Just kidding, but take it to the Chit Chat! Spare my poor thread about six points on a disc! Sad
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