Difference between revisions of "2013 AIME II Problems/Problem 15"
ProbaBillity (talk | contribs) (I am posting Problem 15 (Problem LaTeXed by v_Enhance on the AoPS Forum) along with my solution.) |
ProbaBillity (talk | contribs) (→Solution) |
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By the Law of Sines, we must have <math>CA = \sin{B}</math> and <math>AB = \sin{C}</math>. | By the Law of Sines, we must have <math>CA = \sin{B}</math> and <math>AB = \sin{C}</math>. | ||
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Now let us analyze the given: | Now let us analyze the given: |
Revision as of 13:41, 4 April 2013
\[ ,
,
, and
for which
where
and
are relatively prime and
is not divisible by the square of any prime. Find
.
Solution
Let's draw the triangle. Since the problem only deals with angles, we can go ahead and set one of the sides to a convenient value. Let .
By the Law of Sines, we must have and
.
Now let us analyze the given:
$
Therefore: Similarly,
Note that the desired value is equivalent to
, which is
. All that remains is to use the sine addition formula and, after a few minor computations, we obtain a result of
. Thus, the answer is
.