Difference between revisions of "2013 USAMO Problems/Problem 2"
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| − | For a positive integer plot equally spaced points around a circle. Label one of them , and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of distinct moves available; two from each point. Let count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that for all | + | For a positive integer <math>n\geq 3</math> plot <math>n</math> equally spaced points around a circle. Label one of them <math>A</math>, and place a marker at <math>A</math>. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of <math>2n</math> distinct moves available; two from each point. Let <math>a_n</math> count the number of ways to advance around the circle exactly twice, beginning and ending at <math>A</math>, without repeating a move. Prove that <math>a_{n-1}+a_n=2^n</math> for all <math>n\geq 4</math> |
Revision as of 18:40, 11 May 2013
For a positive integer 
plot 
equally spaced points around a circle. Label one of them 
, and place a marker at . One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of 
distinct moves available; two from each point. Let 
count the number of ways to advance around the circle exactly twice, beginning and ending at , without repeating a move. Prove that 
for all 
