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DarkerLine
ceci n'est pas une |
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 Posted: 19 Nov 2005 08:18:33 pm Post subject: |
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Quote: 8. Express the following sum:
[attachment=366:attachment]
In the form u * Φ(v * w^x) + y * Ψ(z), where Φ and Ψ can each be one of the six functions sin, cos, tan, sec, csc, and cot, and each of u, v, w, x, y, and z are integers such that v and z are positive, x is greater than or equal to 2, and w is a positive prime number that does not divide v. [font="courier"]Answers written in any other form will not be considered correct! (Angles are expressed in radians.)This was one of the harder problems from a math competition I attended today. I don't believe anyone got it; would anyone here happen to have a clue how to solve it?
(I'm sure the paragraph at the bottom is mostly meaningless and confusing - just simplifying the sum is what they're looking for)
Last edited by Guest on 19 Nov 2005 08:21:39 pm; edited 1 time in total |
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elfprince13
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| Posted: 19 Nov 2005 08:26:09 pm Post subject: |
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| Are calculators allowed? |
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DarkerLine
ceci n'est pas une |
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| Posted: 19 Nov 2005 08:30:46 pm Post subject: |
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| Nope. I don't think anything short of a TI-89 would help you; in fact, the TI-89 isn't much help so far either. Remember, you can't simplify it to just a decimal; you have to do the exact answer. |
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elfprince13
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| Posted: 19 Nov 2005 08:39:43 pm Post subject: |
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| Im not sure how they expect you to do that without photographic memory, and insane mental math skills....and having seen the trig tables in rationalized form |
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DarkerLine
ceci n'est pas une |
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| Posted: 19 Nov 2005 08:49:13 pm Post subject: |
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It's not insane mental math; you have paper, pencil, and about five to ten minutes (10 minutes for this and another problem).
I agree that this year's contest was overly hard, in this and other areas, especially when compared to last year's. |
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thornahawk
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Joined: 27 Mar 2005
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| Posted: 20 Nov 2005 09:36:59 am Post subject: |
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I'm feeling lazier than usual today, so I'll just attach part one of my simplification.
:D
thornahawk
P.S. "Some of the deeper truths of the real line are best appreciated in the complex plane." |
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Liazon
title goes here
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Joined: 01 Nov 2005
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| Posted: 20 Nov 2005 12:01:33 pm Post subject: |
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was that from Math Olympiad?
and ole 'it that's frigin' hard!!!
then again i haven't taken trig or precalc |
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DarkerLine
ceci n'est pas une |
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| Posted: 20 Nov 2005 03:06:31 pm Post subject: |
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Nope, Duke Math Meet as stated in the title. (http://www.math.duke.edu/dumu/)
I thought I'd be able to figure it now after seeing the first part of the solution, but I still haven't got it. Grr.. I don't like trig. |
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DigiTan
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| Posted: 20 Nov 2005 10:56:59 pm Post subject: |
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| Those slippery guys really went nuts on their trig identities. :sick: |
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thornahawk
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| Posted: 21 Nov 2005 10:58:06 am Post subject: |
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Aw, bwuddy widdle twig sum biting ya hard? :D
Another hint: if you look at my partial solution carefully, you'd see a good portion of terms cancel out, leaving you with two dangling csc() terms and one (still sort of nasty) csc() series. Start from there. I know you can. ;)
thornahawk |
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DigiTan
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| Posted: 21 Nov 2005 05:45:24 pm Post subject: |
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Well, it wouldn't be so bad if there weren't all those integer rules. I really think they're meant to distract from the actual problem.
Also, what program do you use to write out formulas? |
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DarkerLine
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| Posted: 21 Nov 2005 06:14:51 pm Post subject: |
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| Don't know about thornahawk, I use OpenOffice but I suppose Maple would be easier. |
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alexrudd
pm me if you read this
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| Posted: 21 Nov 2005 08:15:27 pm Post subject: |
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Isn't the answer to those questions always either 1, 0, or -1? :)
I don't know how to evaluate those sigmas, my guess would be it's a series but my trig course skipped sequences and series.  |
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DigiTan
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| Posted: 21 Nov 2005 10:10:51 pm Post subject: |
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Yeah, the sigma's a series. You swap the "i"s for 0, then 1, 2, and so on, and keep a running total of it all until you get to 17 or it snowballs out of control. :sick:
I think I see how it went down. I've gotta say, there's no way I could've gotten that one in 10 minutes. What kind of sicko comes up with these questions?  |
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thornahawk
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| Posted: 21 Nov 2005 11:34:14 pm Post subject: |
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I used to do all my mathematical typesetting in TeX, but I switched to the less bohemian program MathType (the souped up version of Microsoft Equation Editor) from here. :)
DigiTan, I'd be inclined to the belief that the ones who wrote the questions had WAAAY too much time on their hands. :D
You may be on to something, Alex. ;)
thornahawk |
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