Requirements
1. How do i know that everything is correct? Can i make the payment
after everything is verified?
2. All work must be clearly explained and ALL STEPS should be WRITTEN
DOWN (E-mail i mean is ok, scan your work then e-mail it to me).
Everything should be ACCURATE; otherwise no partial payment.
3. It needs to be done by this WEDNESDAY MAY 4, 2005 12:01 AM
4. I just want to challenge people out there, fortunely i have extra money
i want to spend with you guys.
5. Somehow i can't copy and paste the problem to this website, so i
think the best way that i send you the problem through email whenever
you are ready.
6. So the problem is not here, i need to send you.
7. All the best. I hope we can do this.

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Clarification of Question by m4th_g4dg3t-ga on 01 May 2005 15:31 PDT

How am i going to put the questions here because it can't copy the
math characters? Any suggestions how?

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Clarification of Question by m4th_g4dg3t-ga on 01 May 2005 18:27 PDT

I will  try to put it here. Let me know if it's not clear.

1. Let n be a natural number and A1,...,An > 0 pairwise different
(Pairwise different is underline = important) positive real numbers.
Show that if ( here it's the problem i mean the character i can't put
it on the web it's like Y but upside down, it's Lamda's sign)


Show that if Y1 (Lamda 1),..., Yn (Lamda n) are such real numbers that the equality
            Y(Lamda)1e^((A1)(X))+...+Y(Lamda)ne^(An)(X) = 0
holds true for all x Element (E or belong to) R then Y1 (Lamda
1)=...=Yn(Lamda n) =0

#2
Show that there are infinitely many real numbers x in the interval [0,
(pie/2)] such that both sin x and cos x are rational numbers. (Note: A
real number y is called rational if it can be represented as (m/n) for
some integers m and n)

#3
Show that for any (any is underline = important) real number x E
(element or belong to) [-1,1] and any positive real number B > 0 there
exists a natural number^2 (to the power of 2) n such that

                           |sin  ((square root of 2)*pie*n)-x|< B 

In other words, prove that we can make sin((square root of two)*pie*n)
arbitrarily close to x by an appropriate choice of n E (Element or
belong to) N
(Here a little bit -- use the fact that square root of 2 is not a
rational number. A perfect work should also include a proof that
square root of 2 is in fact not a rational number.)

Ok there you go..if you have any questions, let me know ASAP. I will
check my email periodically. Happy Math!!

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Clarification of Question by m4th_g4dg3t-ga on 02 May 2005 20:21 PDT

Here it's the link for the problem

http://www.freefilehosting.net/play.cfm?id=A08C93A6-F822-BF94-3D591557FCA164DB

Let me know if you have any troubles. Happy Math!

Due Date change to May 5,2005 11:00 PM

Email contact between customers and Google Answers Researchers is not
permitted. You'll need to post your problem here or post it elsewhere
on the Web and provide a link.

For better understand try to post a file with a problem here and let
us know the link to see it:
http://www.freefilehosting.net/

A couple of hints:

1) Write 0 as a linear combination of the functions, and then divide
by the exponential with the largest A_1.  Take the limit as x -->
[infty] to see that the coefficient corresponding to this exponential
must be 0.  Now use induction.

2) Use the intermediate value theorem.

Thank you for all the comments.