Difference between revisions of "1996 USAMO Problems/Problem 1"
Zhuangzhuang (talk | contribs) |
Zhuangzhuang (talk | contribs) |
||
Line 1: | Line 1: | ||
− | '''Problem''' | + | '''Problem:''' |
+ | |||
+ | ---- | ||
+ | |||
Prove that the average of the numbers <math> n\sin n^{\circ}\; (n = 2,4,6,\ldots,180) </math> is <math>\cot 1^\circ</math>. | Prove that the average of the numbers <math> n\sin n^{\circ}\; (n = 2,4,6,\ldots,180) </math> is <math>\cot 1^\circ</math>. | ||
− | + | ||
Revision as of 21:51, 4 November 2012
Problem:
Prove that the average of the numbers is
.
Solution:
First, as we omit that term. Now, we multiply by
to get, after using product to sum,
.
This simplifies to
. Since
this simplifies to
. We multiplied by
in the beginning, so we must divide by it now, and thus the sum is just
, so the average is
, as desired.