Difference between revisions of "2015 AMC 12B Problems/Problem 25"
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Multiplying out, we have <math>1008\sqrt{2}+1008\sqrt{6}</math>, so the answer is <math>1008+2+1008+6= \boxed {\bold {(B)}\; 2024}</math>. | Multiplying out, we have <math>1008\sqrt{2}+1008\sqrt{6}</math>, so the answer is <math>1008+2+1008+6= \boxed {\bold {(B)}\; 2024}</math>. | ||
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==Solution 3== | ==Solution 3== |
Revision as of 12:39, 4 February 2018
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
.
Solution 3
We first notice that if the bee is turning 30 degrees each turn, it will take 12 turns to be looking in the same direction when the bee initially left. This means we simply need to answer the question; how far will the bee be when the bee is facing in the same direction?
First we use the fact that after 3 turns, the bee will be facing in a direction perpendicular to the the initial direction. From here we can draw a perpendicular from to the line
intersecting a point
. We will also place the point
at the intersection of
and
. In addition, the point
is placed at the perpendicular dropped from
to the line
. We will also set the distance
and thus
. With this perpendicular we see that the triangle
is a 30-60-90 triangle. This means that the length
and the length
. We can also see that the triangle
is a 30-60-90 triangle and thus
and
. Now if we continue this across all
and set the point
to the coordinates
. As you can see, we are inherently putting a “box” around the figure. Doing similar calculations for all four “sides” of this spiral we get that the length
,
,
,
, and finally
.
Here the point is defined as the intersection of lines
and
. The point
is defined as the intersection of lines
and
. Finally, the point
is defined as the intersection of lines
and
. Note that our spiral stops at
before the next spiral starts. Calculating the offset from the x and the y direction, we see that the offset, or the new point
, is
. This is an interesting property that the points’ coordinate changes by a constant offset no matter what
is. Since the new point’s subscript changes by 12 each time and we see that 2016 is divisible by 12, the point
. Using similar 30-60-90 triangle properties, we see that
. Using the distance formula, the numbers cancel out nicely (1008 is divisible by 168, so take 168 when using the distance formula) and we see that the final answer is
which gives us a final answer of
.
-bowmanrocks32
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.