2005 AIME II Problems/Problem 11

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.

2005 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2005 AIME II Problems/Problem 11

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