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        riddles >> putnam exam (pure math) >> Area Of a Circle
        
(Message started by: johhnywave on Mar 29th, 2007, 10:45pm)
Title: Area Of a Circle
Post by johhnywave on Mar 29th, 2007, 10:45pm
I put it into a picture because I don't feel like re-typing it.

http://img257.imageshack.us/img257/536/riddlemethisxy6.jpg

Direct Link: http://img257.imageshack.us/img257/536/riddlemethisxy6.jpg

Good Luck!
Title: Re: Area Of a Circle
Post by ThudanBlunder on Mar 30th, 2007, 1:22am
If
the area of circle A is 3 times the area of circle B
and
the area of square WXYZ is 6 times the area of circle B
then
the area of square WXYZ is 2 times the area of circle A

We know
the area of square WXYZ is 4/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subpi.gif times the area of circle in square WXYZ

So required ratio = 2*(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subpi.gif/4) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/subpi.gif/2  
Title: Re: Area Of a Circle
Post by srn347 on Sep 2nd, 2007, 9:05pm
Now try it with perimeter.
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 3rd, 2007, 5:43am
You should be able to do it.  ;)
Title: Re: Area Of a Circle
Post by srn347 on Sep 6th, 2007, 4:18pm
Me or the uberpuzzler?
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 7th, 2007, 3:30am

Quote:
Me or the uberpuzzler?

Well since your not an uberpuzzler....
Title: Re: Area Of a Circle
Post by ima1trkpny on Sep 7th, 2007, 2:22pm

on 09/07/07 at 03:30:33, mikedagr8 wrote:
Well since your not an uberpuzzler....

LOL... when he becomes uberpuzzler I will be totally depressed... I have yet to come across anything he hasn't answered with complete bogus. Really srn347, would it be too much to ask for you to show a bit of maturity? Though I may be asking too much from someone who is probably no more than 15 at the most...
By the way Mike, nice avatar!  :)
Title: Re: Area Of a Circle
Post by Aryabhatta on Sep 7th, 2007, 3:50pm

on 09/07/07 at 14:22:38, ima1trkpny wrote:
LOL... when he becomes uberpuzzler I will be totally depressed... I have yet to come across anything he hasn't answered with complete bogus. Really srn347, would it be too much to ask for you to show a bit of maturity? Though I may be asking too much from someone who is probably no more than 15 at the most...
By the way Mike, nice avatar!  :)


Now that you mention it, i thought those were burning buildings... but it actually says mikedagr8. Nice.
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 8th, 2007, 2:14am

on 09/07/07 at 15:50:43, Aryabhatta wrote:
Now that you mention it, i thought those were burning buildings... but it actually says mikedagr8. Nice.




Quote:
By the way Mike, nice avatar!  


Thanks :D, srn347 is 13. Still, in certain religions like Judaism, he would be considered an adult, yet he has never shown it. It amazes me that he thinks he can cope with complex problems and ideas way over the top of his head, most of the time it is over mine aswell. It seems he has very selective hearing/reading. He doesn't even read problems, answers them and then runs away when confronted with insults and complaints.


Quote:
LOL... when he becomes uberpuzzler I will be totally depressed


He will never be an uberpuzzler in the way that I meant it. I won't deserve the title if I make that many posts, I just like to enjoy myself not post for the title. I did at one stage, but I have calmed down now.
Title: Re: Area Of a Circle
Post by Barukh on Sep 8th, 2007, 4:12am

on 09/08/07 at 02:14:57, mikedagr8 wrote:
He will never be an uberpuzzler in the way that I meant it. I won't deserve the title if I make that many posts, I just like to enjoy myself not post for the title. I did at one stage, but I have calmed down now.

You will become an uberpuzzler (by the rules of this forum) in a while, hardly 2 months after you joined it.  ::)
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 8th, 2007, 4:19am

on 09/08/07 at 04:12:03, Barukh wrote:
You will become an uberpuzzler (by the rules of this forum) in a while, hardly 2 months after you joined it.  ::)


Yes, but I don't deserve the title, rank maybe, but title no. I am not an uberpuzzler, If I were, well we can only imagine the possibilities. By this I mean, that 5 stars, maybe, being called an uberpuzzler, not at the moment.

Look at flamingdragon. He was averaging 40 posts every 12 hours, he went from 150~ - 650+ in a few weeks if that.
Title: Re: Area Of a Circle
Post by ThudanBlunder on Sep 8th, 2007, 4:54am

on 09/08/07 at 04:19:08, mikedagr8 wrote:
Look at flamingdragon. He was averaging 40 posts every 12 hours, he went from 150~ - 650+ in a few weeks if that.

And Iceman, whose posts often consist of of only one syllable, yes and no.
Still, at least he contributes many WTF? type puzzles.
But I think I am still the numero uno puzzle poster.  8)  

In terms of amount of lines posted [and helpfulness of posts], I would say that Icarus is #1, probably followed by towr.  
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 8th, 2007, 4:57am

on 09/08/07 at 04:54:00, ThudanBlunder wrote:
And Iceman, whose posts often consist of of only one syllable, yes and no.
Still, at least he contributes many WTF? type puzzles.
But I think I am still the numero uno puzzle poster.  8)  

In terms of amount of lines posted [and helpfulness of posts], I would say that Icarus is #1, probably followed by towr.  



And I would agree with you there. Although since Icarus has been away for several months now, I would bump you up to #2. If he ever returns (let's all hope he does), I would have to move you down.
Title: Re: Area Of a Circle
Post by ThudanBlunder on Sep 8th, 2007, 5:14am

on 09/08/07 at 04:57:21, mikedagr8 wrote:
Although since Icarus has been away for several months now, I would bump you up to #2.

Hmm....Grimbal might have something to say about that.
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 8th, 2007, 5:16am

on 09/08/07 at 05:14:11, ThudanBlunder wrote:
Hmm....Grimbal might have something to say about that.


Equal second with Grimbal then. I mean, at least he has cool pictures everytime the page refreshes.
Title: Re: Area Of a Circle
Post by srn347 on Sep 8th, 2007, 11:53am
Your comments don't even deserve a response, but unfortunetly it takes a response to say that.
Title: Re: Area Of a Circle
Post by towr on Sep 9th, 2007, 7:40am

on 09/07/07 at 14:22:38, ima1trkpny wrote:
LOL... when he becomes uberpuzzler I will be totally depressed...
Well, if it would come to that, I'm sure we could petition William to make a special exception and reset his post count;)
Title: Re: Area Of a Circle
Post by Barukh on Sep 9th, 2007, 10:33pm

on 09/08/07 at 02:14:57, mikedagr8 wrote:
srn347 is 13.

How do you know?
Title: Re: Area Of a Circle
Post by ima1trkpny on Sep 9th, 2007, 10:42pm

on 09/09/07 at 22:33:32, Barukh wrote:
How do you know?

I don't know how mikedagr8 knows, but it makes sense with his math level he just told me was advanced algebra (the equivalent of the U.S. Algebra 2) which would be possible if he was taking honors courses, etc (which would explain the over-confident, cocky attitude...)
Title: Re: Area Of a Circle
Post by towr on Sep 10th, 2007, 12:26am

on 09/09/07 at 22:33:32, Barukh wrote:
How do you know?
srn said that somewhere.
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 10th, 2007, 2:18am
70   riddles / hard / Re: Hard: 24   Aug 29th, 2007, 6:07am  
Started by ootte | Post by srn347  

Quote:
In the future they'll have a way to define it. I'm already in that process and I'm 13.  
It's one of the 24 pages. It amazes me he says now that he can use divsion, yet in the original page he says we cant divide.


Quote:
3/7 involves decimal points which are forbidden. Exponents and sqrts aren't useable either. How about (3-3)/(7-7) or vice-verca. 0/0 is anything so it could be 24.


This guy really is a character. Remember, according to a 13 year old, who says he has done calculus, 0/0 is 24.  ;) I mean, I don't start learning calculus for another year, but he has already learnt it, probably mastered it to his standards as well.
Title: Re: Area Of a Circle
Post by ima1trkpny on Sep 10th, 2007, 1:26pm
Really?  :o Actually Mike I would have figured you had already started it... in fact you probably are already much closer than you think... at least in all the courses I had, you were learning the concepts and building the foundation ahead of time, they just didn't tell you as such. Then when you got to calculus they just went "suprise! now here is how to use it..." But you'll probably like it... it just makes so much sense you wonder why everyone gives it such a stigma. Good luck!  ;D
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 11th, 2007, 12:41am

on 09/10/07 at 13:26:32, ima1trkpny wrote:
Really?  :o Actually Mike I would have figured you had already started it... in fact you probably are already much closer than you think... at least in all the courses I had, you were learning the concepts and building the foundation ahead of time, they just didn't tell you as such. Then when you got to calculus they just went "suprise! now here is how to use it..." But you'll probably like it... it just makes so much sense you wonder why everyone gives it such a stigma. Good luck!  ;D


YES, THAT'S EXACTLY WHAT IS HAPPENING!!!

Next topic we start derivatives....CALCULUS WOOOOH! We're doing circular functions, then on to 'rates of change' and 'differentiation of polynomials'!!! ;D :D  8) :o :)
Title: Re: Area Of a Circle
Post by Sameer on Sep 11th, 2007, 8:55am
Wait how can you start on derivatives without limits? Btw what is the thread about?
Title: Re: Area Of a Circle
Post by mikedagr8 on Sep 11th, 2007, 2:35pm

on 09/11/07 at 08:55:55, Sameer wrote:
Wait how can you start on derivatives without limits? Btw what is the thread about?


Rates of change has a sub-chapter I suppose you would call it, with limits as a topic. It's a jumbled book, and the teacher teaches in his own methods so I guess we will see. This thread was just a general conversation/topic. I'm not 100% sure anymore  :-/
Title: Re: Area Of a Circle
Post by Obob on Sep 11th, 2007, 3:25pm
Well you can do differentiation of polynomials without knowing about limits.  You can't prove your answer is correct (not that you even know what correct means, since you can't define derivative), but at least you can differentiate them.

Strangely enough, differentiation of polynomials is actually an important concept in algebraic geometry, which is a field of math where you can talk about curves and stuff over number systems other than the real or complex numbers, where there is no notion of limits.  You simply define the derivative of the polynomial to be what you would expect it to be in the real or complex case.
Title: Re: Area Of a Circle
Post by Eigenray on Sep 12th, 2007, 1:51am
Here is an example of how differentiation can be useful, even though limits make no sense.

You may know that a polynomial f(x) has repeated roots (over http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif, say) if and only if f(x) and f'(x) have a common (non-constant) factor.  That is, by applying polynomial division with remainder, you can perform the Euclidean algorithm to compute gcd(f, f').  If you get a constant, then f has distinct roots (f is separable).

But in fact, this result holds with polynomials over any field F, if we compute the derivative formally.  That is, if f(x) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifakxk, define f'(x) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifk akxk-1.  The proof only uses that F is a field.

One reason this result is useful is that one can show certain fields F are "perfect", which means any irreducible polynomial over F has no repeated roots, or that any algebraic field extension of F is "separable", which is an important concept in field theory.

In fact, this is almost immediate if F has characteristic 0 (that is, it contains a copy of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbq.gif): if f is a non-constant polynomial, then f' is non-zero, and has smaller degree than f.  If f is irreducible then, it can't have a common factor with f', and so f has distinct roots.

But suppose F=http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbf.gifp is the finite field with p elements, also known as "http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbz.gif mod p", or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbz.gif/(phttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbz.gif) = {0,1,...,p-1}.  Then there are non-constant polynomials whose derivative is 0.  For example, if g(x)=xp-1, then g'(x) = pxp-1 = 0, even though g(x) is non-constant (even as a function on F).  On the other hand, if h(x)=xp-x+1, say, then h(a)=1 for all a http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gif F, so is "constant" in that sense, even though h'(x) = -1.

Now, since g'=0, g and g' share a common factor (namely g itself), so g has repeated roots.  In fact, g(x) = xp-1 = (x-1)p.  But if we were to view g as a polynomial over http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbq.gif, it would be separable.

On the other hand, since h'(x)=-1, it is relatively prime to h, so h has no repeated roots over F.

The ability for f' to be 0 even when f is non-constant makes characteristic p fields more interesting.  Finite fields are still perfect, but many infinite fields aren't.

Derivatives can be generalized even more, to any algebra A over a field F.  An F-linear map d:A->A is called a derivation if d(uv)=d(u)v + ud(v).  For example, if A=F[x], then d is uniquely determined by d(x); if d(x)=1, then d is the usual derivative.  But in general, the set of all derivations on A is itself a vector space over F.  This comes up in differential geometry, Lie algebras, and stuff.  Ramble ramble.
Title: Re: Area Of a Circle
Post by DC1E2L on Oct 23rd, 2007, 4:29am
Officially started calculus, and have learnt a few things inc. limits as you guys said. And a few forms of the notation. Also the basics of derivatives.


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