2015 AIME I Problems/Problem 13
Problem
With all angles measured in degrees, the product , where
and
are integers greater than 1. Find
.
Solution 1
Let . Then from the identity
we deduce that (taking absolute values and noticing
)
But because
is the reciprocal of
and because
, if we let our product be
then
because
is positive in the first and second quadrants. Now, notice that
are the roots of
Hence, we can write
, and so
It is easy to see that
and that our answer is
.
Solution 2
Let
because of the identity
we want
Thus the answer is
Solution 3
Similar to solution 2, so we use and we find that:
Now we can cancel the sines of the multiples of
:
So
and we can apply the double-angle formula again:
Of course,
is missing, so we multiply it to both sides:
Now isolate the product of the sines:
And the product of the squares of the cosecants as asked for by the problem is the square of the inverse of this number:
\[csc^21 csc^23 csc^25 csc^27 \cdots csc^283 csc^285 csc^287 csc^289=(\frac{1}{2^{\frac{89}{2}]})^2=(2^{\frac{89}{2}})^2=2^{89}\] (Error compiling LaTeX. Unknown error_msg)
The answer is therefore .
See Also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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