Mock AIME 6 2006-2007 Problems/Problem 10
Problem
Given a point in the coordinate plane, let
be the
rotation of
around the point
. Let
be the point
and
for all integers
. If
has a
-coordinate of
, what is
?
Solution
Let be the rotational matrix for a point along the origin:
For
Let be the point of rotation, then
Let's write in matrix form as:
, where
and
are the
and
coordinates of
respectively.
We can write the equation of by translating the
to the origin, multiply it by the rotation matrix
and then add the point subtracted:
Now we find :
For this problem, we're only interested in the -coordinate. So,
Notice that since , then
.
$P_{y_0}=0
Since we're looking at a$ (Error compiling LaTeX. Unknown error_msg)y433
433 \equiv 1\; (mod\;2) \not\equiv P_{y_0}\; (mod\;2)$
~Tomas Diaz. orders@tomasdiaz.com