2008 AIME II Problems/Problem 9
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 .
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
If you're curious, the points do eventually form an octagon and repeat. Seems counterintuitive, but believe it or not, it happens.
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
where a is cis. By De-Moivre's theorem, =cis. Therefore,
Furthermore, . Thus, the final answer is
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).
Let . We assume that the rotation matrix here. Then we have
This simplifies to
Since , so we have , giving . The answer is yet .
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