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Aaron the ant

Source: 2015 AMC 10 B Problem 24

February 26, 2015
analytic geometryAMC

Problem Statement

Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin p0=(0,0)p_0=(0,0) facing to the east and walks one unit, arriving at p1=(1,0)p_1=(1,0). For n=1,2,3,n=1,2,3,\dots, right after arriving at the point pnp_n, if Aaron can turn 9090^\circ left and walk one unit to an unvisited point pn+1p_{n+1}, he does that. Otherwise, he walks one unit straight ahead to reach pn+1p_{n+1}. Thus the sequence of points continues p2=(1,1),p3=(0,1),p4=(1,1),p5=(1,0)p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0), and so on in a counterclockwise spiral pattern. What is p2015p_{2015}?
<spanclass=latexbold>(A)</span>(22,13)<spanclass=latexbold>(B)</span>(13,22)<spanclass=latexbold>(C)</span>(13,22)<spanclass=latexbold>(D)</span>(13,22)<spanclass=latexbold>(E)</span>(22,13) <span class='latex-bold'>(A) </span> (-22,-13)\qquad<span class='latex-bold'>(B) </span> (-13,-22)\qquad<span class='latex-bold'>(C) </span> (-13,22)\qquad<span class='latex-bold'>(D) </span> (13,-22)\qquad<span class='latex-bold'>(E) </span> (22,-13)