Trig

Filed on Today, at 8:09 am

1. Prove that for any positive integer $n$ ,
$\cot ^2 \dfrac{\pi}{2n + 1} + \cot^2 \dfrac{2\pi}{2n + 1} + \dots + \cot^2 \dfrac{n \pi}{2n + 1} = \dfrac{n(2n - 1)}{3}.$
2. Solve the following system in positive real numbers:
$\left (x_1 + \dots + x_k \right ) \left (x_k + \dots + x_{10} \right ) = 1$
for $k = 1,\dots,10$ .

Express the solutions in the form $\dfrac{a\sqrt {b} - c\sqrt {d}}{e}$ , where $a,b,c,d,e$ are positive integers, $b,c$ are square-free, and $\gcd (a,c,e) = 1$ .

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3 Comments have been made

Where the hell do you get all of these random problems...

Well, for 1 see this.

Posted Sat Nov 07, 2009 4:53 pm, by math154

Random problems come from random places. Inside joke.

Posted Sat Nov 07, 2009 6:45 pm, by AIME15

For Number 1, Use the Cylindrical Shell Method to generalize about the sum....duh.

Posted Mon Nov 09, 2009 5:37 pm, by BarbieRocks

THIS BLOG DESCRIPTION IS IN ALL CAPS

Blog owner: AIME15
Contributors: serialk11r, james4l, Mewto55555, AceOfDiamonds

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u should start PoTD/W/M again

By BarbieRocks, on Yesterday, at 6:22 pm
lol

By randomguy64, on Yesterday, at 5:55 pm
do you delete all my shouts??????

By BarbieRocks, on Thu Nov 26, 2009 7:12 am
m places. Inside joke.

By PhireKaLk6781, on Sun Nov 08, 2009 10:22 am
Merry Halloween Everybody!

By BarbieRocks, on Sat Oct 31, 2009 7:24 am
Aay. Also known as Aye/Nay depending on the situation. Hahaha.

By randomguy64, on Wed Oct 28, 2009 7:06 pm
aye

By BarbieRocks, on Wed Oct 28, 2009 2:52 pm
everyone who enjoys listening to birds sing say 'aye' without the ''.

By AceOfDiamonds, on Tue Oct 27, 2009 7:45 pm
Wow... way to go with your signature lol

By SonyWii, on Fri Oct 23, 2009 4:16 pm
wow...that was mean

By gauss1181, on Thu Oct 22, 2009 7:36 am