2008 AIME II Problems/Problem 9
Problem
A particle is located on the coordinate plane at . Define a move for the particle as a counterclockwise rotation of
radians about the origin followed by a translation of
units in the positive
-direction. Given that the particle's position after
moves is
, find the greatest integer less than or equal to
.
Solution
Solution 1
Show periodic with steps, then invert twice. Template:Incomplete
Solution 2
Let the particle's position be represented by a complex number. The transformation takes to
where
and
. We let
and
so that we want to find
.
Basically, the thing comes out to
Notice that
Furthermore,
. So our final answer is
$$ (Error compiling LaTeX. Unknown error_msg)
5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}.$
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |