2008 AIME II Problems/Problem 9
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 .
Solution
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 "a" be the inclination of OP to the x-axis. If (x', y') is the position of the particle after a move from P, then and . Let (xn, yn) be the position of the particle after the nth move, where x0 = 5 and y0 = 0. Then , . This implies , . Substituting x0 = 5 and y0 = 0, we have x8 = 5 and y8 = 0 again for the first time. and . Hence the final answer is
Solution 2
Let the particle's position be represented by a complex number. The transformation takes to where and . We let and so that we want to find .
Basically, the thing comes out to
Notice that
Furthermore, . Thus, the final answer is
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 |