Difference between revisions of "2008 AIME II Problems/Problem 9"
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== 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>. | ||
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== Solutions == | == Solutions == | ||
=== Solution 1 === | === Solution 1 === | ||
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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>: | 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>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> | + | <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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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 | + | === 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 | 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> | <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> |
Latest revision as of 22:37, 28 January 2024
Contents
Problem
A particle is located on the coordinate plane at . Define a move for the particle as a counterclockwise rotation of
radians about the origin followed by a translation of
units in the positive
-direction. Given that the particle's position after
moves is
, find the greatest integer less than or equal to
.
Solutions
Solution 1
Let be the position of the particle on the
-plane,
be the length
where
is the origin, and
be the inclination of OP to the x-axis. If
is the position of the particle after a move from
, then we have two equations for
and
:
Let
be the position of the particle after the nth move, where
and
. Then
,
. This implies
,
.
Substituting
and
, we have
and
again for the first time. Thus,
and
. Hence, the final answer is
![$5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}$](http://latex.artofproblemsolving.com/d/6/e/d6e93554c7f003c0766d1509e99aad3fbd7d9f19.png)
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 rotates the object in the complex plane by
counterclockwise. In this case, we use
. Therefore, applying the rotation and shifting the coordinates by 10 in the positive x direction in the complex plane results to
![$a_{150} = (((5a + 10)a + 10)a + 10 \ldots) = 5a^{150} + 10 a^{149} + 10a^{148}+ \ldots + 10$](http://latex.artofproblemsolving.com/b/d/b/bdba935af30d8eca5158fe6587d0c83bf1952d15.png)
where a is cis. By De-Moivre's theorem,
=cis
.
Therefore,
![$10(a^{150} + \ldots + 1)= 10(1 + a + \ldots + a^6) = - 10(a^7) = - 10(\frac{ \sqrt {2} }{2} - \frac{i\sqrt {2}} {2})$](http://latex.artofproblemsolving.com/a/f/e/afe0bcb6dc41f4d96f940d2d64b59865ef9566df.png)
Furthermore, . Thus, the final answer is
![$5\sqrt {2} + 5(\sqrt {2} + 1) \approx 19.1 \Longrightarrow \boxed{019}$](http://latex.artofproblemsolving.com/d/6/e/d6e93554c7f003c0766d1509e99aad3fbd7d9f19.png)
Solution 3
As before, consider as a complex number. Consider the transformation
. This is a clockwise rotation of
by
radians about the points
. Let
denote one move of
. Then
Therefore, rotates along a circle with center
. Since
,
, as desired (the final algebra bash isn't bad).
Solution 4
Let . We assume that the rotation matrix
here. Then we have
![$T^{150}\begin{pmatrix}5\\0\end{pmatrix}=R(R(...R(R\begin{pmatrix}5\\0\end{pmatrix}+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}...)+\begin{pmatrix}10\\0\end{pmatrix})+\begin{pmatrix}10\\0\end{pmatrix}$](http://latex.artofproblemsolving.com/c/7/7/c779a76d8286dbe05981c0790cb34629e9605e30.png)
This simplifies to
![$R^{150}\begin{pmatrix}5\\0\end{pmatrix}+(I+R^2+R^3+...+R^{149})\begin{pmatrix}10\\0\end{pmatrix}$](http://latex.artofproblemsolving.com/c/6/f/c6f6b8137274ebe4c9479bdef91f1dad1f7d7633.png)
Since , so we have
, giving
. The answer is yet
.
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.