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        riddles >> medium >> Broken Calculator
        
(Message started by: FiBsTeR on May 16th, 2008, 3:23pm)
Title: Broken Calculator
Post by FiBsTeR on May 16th, 2008, 3:23pm
A calculator is broken so that the only keys that still work are the sin, cos, tan, arcsin, arccos, and arctan buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Source: USAMO
Title: Re: Broken Calculator
Post by Eigenray on May 18th, 2008, 10:55am
In fact, you can compute [hide]sqrt(q)[/hide] for any q.
Title: Re: Broken Calculator
Post by Barukh on May 18th, 2008, 11:01am

on 05/18/08 at 10:55:28, Eigenray wrote:
In fact, you can compute [hide]sqrt(q)[/hide] for any q.

That's what I arrived at also.

Does that mean the problem at USAMO was invalid?

???
Title: Re: Broken Calculator
Post by FiBsTeR on May 18th, 2008, 11:14am

on 05/18/08 at 11:01:03, Barukh wrote:
Does that mean the problem at USAMO was invalid?


It's not invalid, it's just not asking for the most specific set of numbers that you can attain with this calculator. They may have thought that asking about the square roots of the rationals would have hinted at the proof.

EDIT: The fact that Eigenray hid that part of the statement suggests that this may have been the reason behind it.

Good job in your solutions, by the way.  :)
Title: Re: Broken Calculator
Post by Barukh on May 18th, 2008, 9:45pm
Yes, of course! The queastion in my previous post is a complete nonsense.

Here’s my solution (in short):

[hideb]
I will use short notations S for sine, S-1 for arcsine, and similarly for cosine and tangent functions.

Using the following formulas (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Compositions_of_trig_and_inverse_trig_functions), we obtain the transformations:

TS-1TC-1(z) = 1/z
CT-1(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifz) = 1/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(z+1)

We also have SC-1(0) = 1.

It is then sufficient to use the fact (http://en.wikipedia.org/wiki/Continued_fraction#Finite_continued_fractions) that every rational number can be represented as a finite simple continued fraction.
[/hideb]
Title: Re: Broken Calculator
Post by ThudanBlunder on May 19th, 2008, 5:00pm
Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif to any required precision?
Title: Re: Broken Calculator
Post by FiBsTeR on May 19th, 2008, 5:31pm

on 05/18/08 at 21:45:37, Barukh wrote:
TS-1TC-1(z) = 1/z


I'm assuming by your notation that I apply arccos to z first? But then how do you know that z is in the domain of arccos?


on 05/18/08 at 21:45:37, Barukh wrote:
We also have SC-1(0) = 1.


C(0)=1, as well.  ;)
Title: Re: Broken Calculator
Post by Barukh on May 20th, 2008, 6:31am

on 05/19/08 at 17:31:32, FiBsTeR wrote:
I'm assuming by your notation that I apply arccos to z first? But then how do you know that z is in the domain of arccos?

Not only that, the transformation is not correct!

The correct one IMHO is TS-1CT-1.
Title: Re: Broken Calculator
Post by Barukh on May 20th, 2008, 6:40am

on 05/19/08 at 17:00:22, ThudanBlunder wrote:
Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif to any required precision?

[hide](69)[/hide]?
Title: Re: Broken Calculator
Post by ThudanBlunder on May 20th, 2008, 8:27am

on 05/20/08 at 06:40:13, Barukh wrote:
[hide](69)[/hide]?

???
Title: Re: Broken Calculator
Post by Barukh on May 20th, 2008, 10:46am

on 05/20/08 at 08:27:12, ThudanBlunder wrote:
???

[hide]Vieta[/hide]!
Title: Re: Broken Calculator
Post by ThudanBlunder on May 20th, 2008, 11:00am

on 05/20/08 at 06:40:13, Barukh wrote:
[hide](69)[/hide]?



on 05/20/08 at 10:46:46, Barukh wrote:
[hide]Vieta[/hide]!

I knew you were well-informed, Barukh, but surely you are not also familiar with Viete's sexual proclivities?   ;D
Title: Re: Broken Calculator
Post by Barukh on May 22nd, 2008, 2:14am

on 05/19/08 at 17:00:22, ThudanBlunder wrote:
Using only the addition, multiplication, division, square root, and '2' key (plus Store, Recall, and Sum), how can you approximate http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif to any required precision?

BTW, according to the following source (http://mathworld.wolfram.com/PiFormulas.html), formula (69), division is not needed.
Title: Re: Broken Calculator
Post by FiBsTeR on May 23rd, 2008, 8:34am
My original thought was using (68) in your link, but I couldn't do it without using a second store variable.
Title: Re: Broken Calculator
Post by ThudanBlunder on May 23rd, 2008, 8:54am

on 05/23/08 at 08:34:02, FiBsTeR wrote:
My original thought was using (68) in your link, but I couldn't do it without using a second store variable.

That was the formula I had in mind. It, together with the required iteration, can be found here (http://press.princeton.edu/books/maor/chapter_11.pdf) on page 140.


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