2015 AMC 12B Problems/Problem 25
Contents
Problem
A bee starts flying from point . She flies
inch due east to point
. For
, once the bee reaches point
, she turns
counterclockwise and then flies
inches straight to point
. When the bee reaches
she is exactly
inches away from
, where
,
,
and
are positive integers and
and
are not divisible by the square of any prime. What is
?
Solution 1
Let , a
counterclockwise rotation centered at the origin. Notice that
on the complex plane is:
We need to find the magnitude of on the complex plane. This is an arithmetic/geometric series.
We want to find . First, note that
because
. Therefore
Hence, since , we have
Now we just have to find . This can just be computed directly:
Therefore
Thus the answer is
Solution 2
Here is an alternate solution that does not use complex numbers:
We will calculate the distance from to
using the Pythagorean theorem. Assume
lies at the origin, so we will calculate the distance to
by calculating the distance traveled in the x-direction and the distance traveled in the y-direction. We can calculate this by summing each movement:
A movement of units at
degrees is the same thing as a movement of
units at
degrees, so we can adjust all the cosines with arguments greater than 180 as follows:
Now we group terms with like-cosines and factor out the cosines:
Each sum in the parentheses has 336 terms (except the very last one, which has 335), so by pairing each term, we can see that there are pairs of
. So each sum evaluates to
, except the very last sum, which has 167 pairs of
and an extra 2010, so it evaluates to
. Plugging in these values:
Now that we have how far was traveled in the x-direction, we need to find how far was traveled in the y-direction. Using the same logic as above, we arrive at the sum:
The last step is to use the Pythagorean to find the distance from . This distance is given by:
Multiplying out, we have , so the answer is
.
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
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