Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 15"
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| − | + | ==Problem== | |
| + | Let <math>\Omega</math> denote the value of the sum | ||
<math>\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)</math> | <math>\sum_{k=1}^{40} \cos^{-1}\left(\frac{k^2 + k + 1}{\sqrt{k^4 + 2k^3 + 3k^2 + 2k + 2}}\right)</math> | ||
The value of <math>\tan\left(\Omega\right)</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | The value of <math>\tan\left(\Omega\right)</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Compute <math>m + n</math>. | ||
| + | |||
| + | ==Solution== | ||
| + | {{solution}} | ||
| + | |||
| + | ==See also== | ||
Revision as of 07:29, 14 February 2008
Problem
Let 
denote the value of the sum
The value of 
can be expressed as 
, where 
and 
are relatively prime positive integers. Compute 
.
Solution
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