Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 10"
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<math>P_r=\begin{pmatrix} 2000-k \\ k \end{pmatrix}</math> | <math>P_r=\begin{pmatrix} 2000-k \\ k \end{pmatrix}</math> | ||
+ | |||
+ | Let's write <math>P_n</math> in matrix form as: | ||
+ | |||
+ | <math>P_n=\begin{pmatrix} P_{x_n} \\ P_{y_n} \end{pmatrix}</math> | ||
<math>P_{n+1}=R(P_n-P_r)+P_r=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} P_{x_n}-(2000-k) \\ P_{y_n}-k \end{pmatrix}+\begin{pmatrix} 2000-k \\ k \end{pmatrix}</math> | <math>P_{n+1}=R(P_n-P_r)+P_r=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} P_{x_n}-(2000-k) \\ P_{y_n}-k \end{pmatrix}+\begin{pmatrix} 2000-k \\ k \end{pmatrix}</math> |
Revision as of 15:08, 25 November 2023
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.
Let's write in matrix form as:
~Tomas Diaz. orders@tomasdiaz.com