Japanese Mathematical Olympiad 1995

Geometry Level 4

The sequence {a1,a2,a3,}\{a_1, a_2, a_3, \dots \} is defined by a2n+1=(1)na_{2n + 1} = (-1)^n and a2n=ana_{2n} = a_n. A point PP moves on the coordinate plane as follows.

  • First PP moves from the origin P0(0,0)P_0(0,0) to P1(1,0)P_1 (1,0).
  • After PP has moved to PiP_i, it turns 9090^{\circ} to the left and moves forward 11 unit if ai=1a_i = 1, and turns 9090^{\circ} to the right and moves forward 11 unit if ai=1a_i = -1. Denote this point by Pi+1P_{i + 1}.

Can the point retrace an edge? That is, can Pu=PwP_u = P_w and Pu+1=Pw+1P_{u + 1} = P_{w + 1} for some integers u,wu,w?

×

Problem Loading...

Note Loading...

Set Loading...