Difference between revisions of "2013 USAMO Problems/Problem 2"

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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>
 
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>
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Revision as of 16:59, 3 July 2013

For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. 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 $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1}+a_n=2^n$ for all $n\geq 4$ The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png