Difference between revisions of "2005 AIME II Problems/Problem 11"
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== Problem == | == Problem == | ||
| − | Let <math> | + | Let <math>m </math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of integers such that <math>a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math>m. </math> |
| − | ''Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term <math> | + | ''Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term <math>a_2 </math>, for example, is obviously not integral.'' |
== Solution == | == Solution == | ||
Revision as of 16:21, 16 August 2011
Problem
Let 
be a positive integer, and let 
be a sequence of integers such that 
and 
for 
Find 
Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term 
, for example, is obviously not integral.
Solution
For 
, we have
.
Thus the product 
is a monovariant: it decreases by 3 each time 
increases by 1. Since for 
we have 
, so when 
, will be zero for the first time, which implies that 
, our answer.
See also
| 2005 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
