1999 AIME Problems/Problem 11
Problem
Given that where angles are measured in degrees, and and are relatively prime positive integers that satisfy find
Solution
Let . We could try to manipulate this sum by wrapping the terms around (since the first half is equal to the second half), but it quickly becomes apparent that this way is difficult to pull off. Instead, we look to telescope the sum. Using the identity , we can rewrite as
This telescopes to Manipulating this to use the identity , we get and our answer is .
Alternate Solution
We note that . We thus have that The desired answer is thus .
See also
1999 AIME (Problems • Answer Key • Resources) | ||
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Followed by Problem 12 | |
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