Question 9
A particle is located on the coordinate plane at $(5, 0).$ Define a *move* for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ 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 $|p| + |q|.$