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Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 10"
Line 39: Line 39:
 
<math>P_{y_m} \equiv 1\; (mod\;2)\equiv P_{y_1}\; (mod\;2)</math>  
 
<math>P_{y_m} \equiv 1\; (mod\;2)\equiv P_{y_1}\; (mod\;2)</math>  
  
−
So, we use the recursive formula for <math>P_{y_{n+2}}</math> starting with <math>P_{x_n}</math> as:
+
So, we use the recursive formula for <math>P_{y_{n+2}}</math> starting with <math>P_{y_1}</math>:
  
 +
<math>P_{y_3}=-P_{y_1}+(2)(1)+2</math>
 +
 +
<math>P_{y_5}=-P_{y_3}+(2)(3)+2</math>
 +
 +
<math>P_{y_7}=-P_{y_5}+(2)(5)+2</math>
 +
 +
and we can find <math>P_{y_{2k+1}}</math> for <math>k \in \mathbb{Z}^{+}</math> as:
 +
 +
<math>P_{y_{2k+1}}=-P_{y_{2k-1}}+2(2k-1)+2=-P_{y_{2k-1}}+4k</math>
  
  
 
~Tomas Diaz. orders@tomasdiaz.com
 
~Tomas Diaz. orders@tomasdiaz.com

Revision as of 14:41, 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:

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

For $\theta = 90^\circ$

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

Let $P_r$ be the point of rotation, then $P_r=\begin{bmatrix} 2000-k \\ k \end{bmatrix}$

Let's write $P_n$ in matrix form as: $P_n=\begin{bmatrix} P_{x_n} \\ P_{y_n} \end{bmatrix}$, where $P_{x_n}$ and $P_{y_n}$ are the $x$ and $y$ coordinates of $P_n$ respectively.

We can write the equation of $P_{n+1}$ by translating the $P_n$ to the origin, multiply it by the rotation matrix $R$ and then add the point subtracted:

$P_{n+1}=R(P_n-P_r)+P_r=\begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}\begin{bmatrix} P_{x_n}-(2000-k) \\ P_{y_n}-k \end{bmatrix}+\begin{bmatrix} 2000-k \\ k \end{bmatrix}=\begin{bmatrix} 2000-P_{y_n} \\ P_{x_n}+2n-2000 \end{bmatrix}$

Now we find $P_{n+2}$:

$P_{n+2}=\begin{bmatrix} 2000-P_{y_{n+1}} \\ P_{x_{n+1}}+2n-2000 \end{bmatrix}=\begin{bmatrix} 2000-(P_{x_n}+2n-2000) \\ (2000-P_{y_n})+2(n+1)-2000 \end{bmatrix}=\begin{bmatrix} 4000-P_{x_n}-2n \\ -P_{y_n}+2(n+1) \end{bmatrix}$

For this problem, we're only interested in the $y$-coordinate. So,

$P_{y_{n+1}}=P_{x_n}+2n-2000$

$P_{y_{n+2}}=-P_{y_n}+2(n+1)$

Notice that since $2(n+1) \equiv 0\; (mod\;2)$, then $P_{y_{n+2}} \equiv P_{y_n}\; (mod\;2)$.

Since $P_{y_0}=0$ and we're looking at a $y$-coordinate of $433$, we notice that $433 \equiv 1\; (mod\;2) \not\equiv P_{y_0}\; (mod\;2)$.

Therefore we need to calculate $P_{y_1}$:

$P_{y_1}=P_{x_0}+(2)(0)-2000=2007+2-2000=7$

$P_{y_m} \equiv 1\; (mod\;2)\equiv P_{y_1}\; (mod\;2)$

So, we use the recursive formula for $P_{y_{n+2}}$ starting with $P_{y_1}$:

$P_{y_3}=-P_{y_1}+(2)(1)+2$

$P_{y_5}=-P_{y_3}+(2)(3)+2$

$P_{y_7}=-P_{y_5}+(2)(5)+2$

and we can find $P_{y_{2k+1}}$ for $k \in \mathbb{Z}^{+}$ as:

$P_{y_{2k+1}}=-P_{y_{2k-1}}+2(2k-1)+2=-P_{y_{2k-1}}+4k$


~Tomas Diaz. orders@tomasdiaz.com

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