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Mock AIME 6 2006-2007 Problems/Problem 10

Revision as of 14:03, 25 November 2023 by Tomasdiaz (talk | contribs)

Problem

Given a point $P$ in the coordinate plane, let $T_k(P)$ be the $90^\circ$ rotation of $P$ around the point $(2000-k,k)$. Let $P_0$ be the point $(2007,0)$ and $P_{n+1}=T_n(P_n)$ for all integers $n\ge 0$. If $P_m$ has a $y$-coordinate of $433$, what is $m$?

Solution

Let $R$ be the rotational matrix for a point along the origin:

$R=\begin{pmatrix} cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta) \end{pmatrix}$

For $\theta = 90^\circ$

$R=\begin{pmatrix} cos(90^\circ) & -sin(90^\circ)\\ sin(90^\circ) & cos(90^\circ) \end{pmatrix}=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}$

Let $P_r$ be the point of rotation.

$P_r=\begin{pmatrix} 2000-k \\ k \end{pmatrix}$

$P_{n+1}=R$


~Tomas Diaz. orders@tomasdiaz.com

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