2008 AIME II Problems/Problem 9
From AoPSWiki
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
.
Contents |
Solution
Solution 1
Show periodic with
steps, then invert twice. This solution is incomplete. You can help us out by completing it.
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,
. Thus, the final answer is

See also
| 2008 AIME II (Problems • Resources) | ||
| Preceded by Problem 8 | Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||




