Difference between revisions of "2008 AMC 12A Problems/Problem 25"
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Therefore, <math>a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \frac{1}{2^{98}} \Rightarrow D</math>. | Therefore, <math>a_1 + b_1 = \frac{1}{2^{97}} - \frac{1}{2^{98}} = \frac{1}{2^{98}} \Rightarrow D</math>. | ||
− | Shortcut: no answer has <math>3</math> in the denominator. So the point cannot have orientation <math>(2,4)</math> or <math>(-2,-4)</math>. Also there are no negative answers. Any other non-multiple of <math>90^\circ</math> rotation of <math>30n^\circ</math> would result in the need of radicals. So either it has orientation <math>(4,-2)</math> or <math>(-4,2)</math>. Both answers add up to <math>2</math>. Thus, <math>2/2^99=\boxed{\textbf{(D) }\frac{1}{2^98}}</math>. | + | Shortcut: no answer has <math>3</math> in the denominator. So the point cannot have orientation <math>(2,4)</math> or <math>(-2,-4)</math>. Also there are no negative answers. Any other non-multiple of <math>90^\circ</math> rotation of <math>30n^\circ</math> would result in the need of radicals. So either it has orientation <math>(4,-2)</math> or <math>(-4,2)</math>. Both answers add up to <math>2</math>. Thus, <math>2/2^99=\boxed{\textbf{(D) }\frac{1}{2^{98}}}</math>. |
==See Also== | ==See Also== | ||
{{AMC12 box|year=2008|ab=A|num-b=24|after=Last question}} | {{AMC12 box|year=2008|ab=A|num-b=24|after=Last question}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:40, 13 February 2016
Problem
A sequence ,
,
,
of points in the coordinate plane satisfies
for
.
Suppose that . What is
?
Solution
This sequence can also be expressed using matrix multiplication as follows:
.
Thus, is formed by rotating
counter-clockwise about the origin by
and dilating the point's position with respect to the origin by a factor of
.
So, starting with and performing the above operations
times in reverse yields
.
Rotating clockwise by
yields
. A dilation by a factor of
yields the point
.
Therefore, .
Shortcut: no answer has in the denominator. So the point cannot have orientation
or
. Also there are no negative answers. Any other non-multiple of
rotation of
would result in the need of radicals. So either it has orientation
or
. Both answers add up to
. Thus,
.
See Also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last question |
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 |
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