Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 10"

Revision as of 15:01, 25 November 2023

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 $\theta = 90^\circ$

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

~Tomas Diaz. orders@tomasdiaz.com

@@ Line 3: / Line 3: @@
 ==Solution==
-{{Solution}}
+Let <math>R</math> be the rotational matrix for a point along the origin:
+<math>R=\begin{pmatrix} cos(\theta) & -sin(\theta)\\ sin(\theta) & cos(\theta) \end{pmatrix}</math>
+For <math>\theta = 90^\circ</math>
+<math>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}</math>
+~Tomas Diaz. orders@tomasdiaz.com

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Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 10"

Revision as of 15:01, 25 November 2023

Problem

Solution