Difference between revisions of "2015 AMC 10B Problems/Problem 24"
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Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin <math>p_0=(0,0)</math> facing to the east and walks one unit, arriving at <math>p_1=(1,0)</math>. For <math>n=1,2,3,\dots</math>, right after arriving at the point <math>p_n</math>, if Aaron can turn <math>90^\circ</math> left and walk one unit to an unvisited point <math>p_{n+1}</math>, he does that. Otherwise, he walks one unit straight ahead to reach <math>p_{n+1}</math>. Thus the sequence of points continues <math>p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)</math>, and so on in a counterclockwise spiral pattern. What is <math>p_{2015}</math>? | Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin <math>p_0=(0,0)</math> facing to the east and walks one unit, arriving at <math>p_1=(1,0)</math>. For <math>n=1,2,3,\dots</math>, right after arriving at the point <math>p_n</math>, if Aaron can turn <math>90^\circ</math> left and walk one unit to an unvisited point <math>p_{n+1}</math>, he does that. Otherwise, he walks one unit straight ahead to reach <math>p_{n+1}</math>. Thus the sequence of points continues <math>p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)</math>, and so on in a counterclockwise spiral pattern. What is <math>p_{2015}</math>? | ||
<math> \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) </math> | <math> \textbf{(A) } (-22,-13)\qquad\textbf{(B) } (-13,-22)\qquad\textbf{(C) } (-13,22)\qquad\textbf{(D) } (13,-22)\qquad\textbf{(E) } (22,-13) </math> | ||
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==See Also== | ==See Also== | ||
{{AMC10 box|year=2015|ab=B|num-b=19|num-a=21}} | {{AMC10 box|year=2015|ab=B|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 22:31, 17 March 2015
Problem:
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin 
facing to the east and walks one unit, arriving at 
. For 
, right after arriving at the point 
, if Aaron can turn 
left and walk one unit to an unvisited point 
, he does that. Otherwise, he walks one unit straight ahead to reach . Thus the sequence of points continues 
, and so on in a counterclockwise spiral pattern. What is 
?
Solution:
See Also
| 2015 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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