Difference between revisions of "2005 AIME II Problems/Problem 11"

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Thus the product <math> \displaystyle a_{k}a_{k+1}</math> is a [[monovariant]]: it decreases by 3 each time <math>\displaystyle k</math> increases by 1.  Since for <math>\displaystyle k = 0</math> we have <math>a_{k}a_{k+1} = 37\cdot 72</math>, so when <math>k = \frac{37 \cdot 72}{3} = 888</math>, <math> \displaystyle a_{k}a_{k+1}</math> will be zero for the first time, which implies that <math>\displaystyle m = 889</math>, our answer.
 
Thus the product <math> \displaystyle a_{k}a_{k+1}</math> is a [[monovariant]]: it decreases by 3 each time <math>\displaystyle k</math> increases by 1.  Since for <math>\displaystyle k = 0</math> we have <math>a_{k}a_{k+1} = 37\cdot 72</math>, so when <math>k = \frac{37 \cdot 72}{3} = 888</math>, <math> \displaystyle a_{k}a_{k+1}</math> will be zero for the first time, which implies that <math>\displaystyle m = 889</math>, our answer.
  
== See Also ==
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== See also ==
 
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{{AIME box|year=2005|n=II|num-b=10|num-a=12}}
* [[2005 AIME II Problems/Problem 10| Previous problem]]
 
* [[2005 AIME II Problems/Problem 12| Next problem]]
 
* [[2005 AIME II Problems]]
 
  
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]

Revision as of 16:55, 22 March 2007

Problem

Let $\displaystyle m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of integers such that $\displaystyle a_0 = 37, a_1 = 72, a_m = 0,$ and $a_{k+1} = a_{k-1} - \frac 3{a_k}$ for $k = 1,2,\ldots, m-1.$ Find $\displaystyle m.$

Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term $\displaystyle a_2$, for example, is obviouly not integral.

Solution

For $\displaystyle 0 < k < m$, we have

$\displaystyle a_{k}a_{k+1} = a_{k-1}a_{k} - 3$.

Thus the product $\displaystyle a_{k}a_{k+1}$ is a monovariant: it decreases by 3 each time $\displaystyle k$ increases by 1. Since for $\displaystyle k = 0$ we have $a_{k}a_{k+1} = 37\cdot 72$, so when $k = \frac{37 \cdot 72}{3} = 888$, $\displaystyle a_{k}a_{k+1}$ will be zero for the first time, which implies that $\displaystyle m = 889$, our answer.

See also

2005 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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All AIME Problems and Solutions
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