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2008 AIME II Problems/Problem 9: Difference between revisions

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__TOC__
== Problem ==
== Problem ==
A particle is located on the coordinate plane at <math>(5,0)</math>. Define a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the greatest integer less than or equal to <math>|p| + |q|</math>.
A particle is located on the coordinate plane at <math>(5,0)</math>. Define a ''move'' for the particle as a counterclockwise rotation of <math>\pi/4</math> radians about the origin followed by a translation of <math>10</math> units in the positive <math>x</math>-direction. Given that the particle's position after <math>150</math> moves is <math>(p,q)</math>, find the greatest integer less than or equal to <math>|p| + |q|</math>.


__TOC__
== Solutions ==  
== Solution ==  
=== Solution 1 ===
=== Solution 1 ===


Let P(x, y) be the position of the particle on the xy-plane, r be the length OP where O is the origin, and <math>\theta</math> be the inclination of OP to the x-axis. If (x', y') is the position of the particle after a move from P, then <math>x'=rcos(\pi/4+\theta)+10 = \sqrt{2}(x - y)/2 + 10</math> and <math>y' = rsin(\pi/4+\theta) = \sqrt{2}(x + y)/2</math>.
Let <math>P(x, y)</math> be the position of the particle on the <math>xy</math>-plane, <math>r</math> be the length <math>OP</math> where <math>O</math> is the origin, and <math>\theta</math> be the inclination of OP to the x-axis. If <math>(x', y')</math> is the position of the particle after a move from <math>P</math>, then we have two equations for <math>x'</math> and <math>y'</math>:
<cmath>x'=r\cos(\pi/4+\theta)+10 = \frac{\sqrt{2}(x - y)}{2} + 10</cmath>
<cmath>y' = r\sin(\pi/4+\theta) = \frac{\sqrt{2}(x + y)}{2}.</cmath>
Let <math>(x_n, y_n)</math> be the position of the particle after the nth move, where <math>x_0 = 5</math> and <math>y_0 = 0</math>. Then <math>x_{n+1} + y_{n+1} =  \sqrt{2}x_n+10</math>, <math>x_{n+1} - y_{n+1} = -\sqrt{2}y_n+10</math>. This implies
Let <math>(x_n, y_n)</math> be the position of the particle after the nth move, where <math>x_0 = 5</math> and <math>y_0 = 0</math>. Then <math>x_{n+1} + y_{n+1} =  \sqrt{2}x_n+10</math>, <math>x_{n+1} - y_{n+1} = -\sqrt{2}y_n+10</math>. This implies
<math>x_{n+2} = -y_n + 5\sqrt{2}+ 10</math>, <math>y_{n+2}=x_n + 5\sqrt{2}</math>.
<math>x_{n+2} = -y_n + 5\sqrt{2}+ 10</math>, <math>y_{n+2}=x_n + 5\sqrt{2}</math>.
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<center><math>5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}</math></center>
<center><math>5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}</math></center>


If you're curious, [url="https://www.desmos.com/calculator/febtiheosz"]the points[/url] do eventually form an octagon and repeat. Seems counterintuitive, but believe it or not, it happens.
If you're curious, the points do eventually form an octagon and repeat. Seems counterintuitive, but believe it or not, it happens.
 
https://www.desmos.com/calculator/febtiheosz


=== Solution 2 ===
=== Solution 2 ===
Let the particle's position be represented by a complex number. Recall that multiplying a number by where a is cis<math>\left( \theta \right)</math>. rotates the object in the complex plane by <math>\theta</math> counterclockwise. Therefore, applying the rotation and shifting the coordinates by 10 in the positive x direction in the complex plane results to
Let the particle's position be represented by a complex number. Recall that multiplying a number by cis<math>\left( \theta \right)</math> rotates the object in the complex plane by <math>\theta</math> counterclockwise. In this case, we use <math>a = cis(\frac{\pi}{4})</math>. Therefore, applying the rotation and shifting the coordinates by 10 in the positive x direction in the complex plane results to
<center><math>a_{150} = (((5a + 10)a + 10)a + 10 \ldots) = 5a^{150} + 10 a^{149} + 10a^{148}+ \ldots + 10</math></center>
<center><math>a_{150} = (((5a + 10)a + 10)a + 10 \ldots) = 5a^{150} + 10 a^{149} + 10a^{148}+ \ldots + 10</math></center>
where a is cis<math>\left( \theta \right)</math>. By De-Moivre's theorem, <math>\left(cis( \theta \right)^n )</math>=cis<math>\left(n \theta \right)</math>.
where a is cis<math>\left( \theta \right)</math>. By De-Moivre's theorem, <math>\left(cis( \theta \right)^n )</math>=cis<math>\left(n \theta \right)</math>.
Therefore,
Therefore,
<center><math>10(a^{150} + \ldots + 1) = 10(1 + a + \ldots + a^6) = - 10(a^7) = - 10(\frac{ \sqrt {2} }{2} - \frac{i\sqrt {2}} {2})</math></center>
<center><math>10(a^{150} + \ldots + 1)= 10(1 + a + \ldots + a^6) = - 10(a^7) = - 10(\frac{ \sqrt {2} }{2} - \frac{i\sqrt {2}} {2})</math></center>
Furthermore, <math>5a^{150} = - 5i</math>. Thus, the final answer is
Furthermore, <math>5a^{150} = - 5i</math>. Thus, the final answer is
<center><math>5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}</math></center>
<center><math>5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}</math></center>
==== Solution 3 ====
As before, consider <math>z</math> as a complex number.  Consider the transformation <math>z \to (z-\omega)e^{i\theta} + \omega</math>.  This is a clockwise rotation of <math>z</math> by <math>\theta</math> radians about the points <math>\omega</math>.  Let <math>f(z)</math> denote one move of <math>z</math>.  Then
[[File:2008AIMEII9Sol3.png|center|300px]]
Therefore, <math>z</math> rotates along a circle with center <math>\omega = 5+(5+5\sqrt2)i</math>.  Since <math>8 \cdot \frac{\pi}{4} = 2\pi</math>, <math>f^9(z) = f(z) \implies f^{150}(z) = f^6(z) \implies p+q = \boxed{019}</math>, as desired (the final algebra bash isn't bad).
=== Solution 4 ===
Let <math>T:\begin{pmatrix}x\\y\end{pmatrix}\rightarrow R(\frac{\pi}{4})\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix}</math>. We assume that the rotation matrix <math>R(\frac{\pi}{4}) = R</math> here. Then we have
<center><math>T^{150}\begin{pmatrix}5\\0\end{pmatrix}=R(R(...R(R\begin{pmatrix}5\\0\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}...)+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}</math></center>
This simplifies to
<center><math>R^{150}\begin{pmatrix}5\\0\end{pmatrix}+(I+R^2+R^3+...+R^{149})\begin{pmatrix}10\\0\end{pmatrix}</math></center>
Since <math>R+R^{7}=O, R^2+R^6=O, R^3+R^5=O, I+R^4=O</math>, so we have <math>R^6\begin{pmatrix}5\\0\end{pmatrix}+(-R^6-R^7)\begin{pmatrix}10\\0\end{pmatrix}</math>, giving <math>p=-5\sqrt{2}, q=5\sqrt{2}+5</math>. The answer is yet <math>\lfloor10\sqrt{2}+5\rfloor=\boxed{019}</math>.


== See also ==
== See also ==

Latest revision as of 21:37, 28 January 2024

Problem

A particle is located on the coordinate plane at $(5,0)$. Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Given that the particle's position after $150$ moves is $(p,q)$, find the greatest integer less than or equal to $|p| + |q|$.

Solutions

Solution 1

Let $P(x, y)$ be the position of the particle on the $xy$-plane, $r$ be the length $OP$ where $O$ is the origin, and $\theta$ be the inclination of OP to the x-axis. If $(x', y')$ is the position of the particle after a move from $P$, then we have two equations for $x'$ and $y'$: \[x'=r\cos(\pi/4+\theta)+10 = \frac{\sqrt{2}(x - y)}{2} + 10\] \[y' = r\sin(\pi/4+\theta) = \frac{\sqrt{2}(x + y)}{2}.\] Let $(x_n, y_n)$ be the position of the particle after the nth move, where $x_0 = 5$ and $y_0 = 0$. Then $x_{n+1} + y_{n+1} = \sqrt{2}x_n+10$, $x_{n+1} - y_{n+1} = -\sqrt{2}y_n+10$. This implies $x_{n+2} = -y_n + 5\sqrt{2}+ 10$, $y_{n+2}=x_n + 5\sqrt{2}$. Substituting $x_0 = 5$ and $y_0 = 0$, we have $x_8 = 5$ and $y_8 = 0$ again for the first time. Thus, $p = x_{150} = x_6 = -5\sqrt{2}$ and $q = y_{150} = y_6 = 5 + 5\sqrt{2}$. Hence, the final answer is

$5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}$

If you're curious, the points do eventually form an octagon and repeat. Seems counterintuitive, but believe it or not, it happens.

https://www.desmos.com/calculator/febtiheosz

Solution 2

Let the particle's position be represented by a complex number. Recall that multiplying a number by cis$\left( \theta \right)$ rotates the object in the complex plane by $\theta$ counterclockwise. In this case, we use $a = cis(\frac{\pi}{4})$. Therefore, applying the rotation and shifting the coordinates by 10 in the positive x direction in the complex plane results to

$a_{150} = (((5a + 10)a + 10)a + 10 \ldots) = 5a^{150} + 10 a^{149} + 10a^{148}+ \ldots + 10$

where a is cis$\left( \theta \right)$. By De-Moivre's theorem, $\left(cis( \theta \right)^n )$=cis$\left(n \theta \right)$. Therefore,

$10(a^{150} + \ldots + 1)= 10(1 + a + \ldots + a^6) = - 10(a^7) = - 10(\frac{ \sqrt {2} }{2} - \frac{i\sqrt {2}} {2})$

Furthermore, $5a^{150} = - 5i$. Thus, the final answer is

$5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}$

Solution 3

As before, consider $z$ as a complex number. Consider the transformation $z \to (z-\omega)e^{i\theta} + \omega$. This is a clockwise rotation of $z$ by $\theta$ radians about the points $\omega$. Let $f(z)$ denote one move of $z$. Then

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Therefore, $z$ rotates along a circle with center $\omega = 5+(5+5\sqrt2)i$. Since $8 \cdot \frac{\pi}{4} = 2\pi$, $f^9(z) = f(z) \implies f^{150}(z) = f^6(z) \implies p+q = \boxed{019}$, as desired (the final algebra bash isn't bad).

Solution 4

Let $T:\begin{pmatrix}x\\y\end{pmatrix}\rightarrow R(\frac{\pi}{4})\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix}$. We assume that the rotation matrix $R(\frac{\pi}{4}) = R$ here. Then we have

$T^{150}\begin{pmatrix}5\\0\end{pmatrix}=R(R(...R(R\begin{pmatrix}5\\0\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}...)+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}$

This simplifies to

$R^{150}\begin{pmatrix}5\\0\end{pmatrix}+(I+R^2+R^3+...+R^{149})\begin{pmatrix}10\\0\end{pmatrix}$

Since $R+R^{7}=O, R^2+R^6=O, R^3+R^5=O, I+R^4=O$, so we have $R^6\begin{pmatrix}5\\0\end{pmatrix}+(-R^6-R^7)\begin{pmatrix}10\\0\end{pmatrix}$, giving $p=-5\sqrt{2}, q=5\sqrt{2}+5$. The answer is yet $\lfloor10\sqrt{2}+5\rfloor=\boxed{019}$.

See also

2008 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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