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2002 OIM Problems/Problem 5

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Problem

The sequence of real numbers $a1, a2, \cdots$ is defined as:

\[a_1 = 56, a_{n+1} = a_n - \frac{1}{a_n}\]

for every integer $n \ge 1$.

Prove that there exists an integer $k$, $1 \le k \le 2002$, such that $a_k < 0$.

~translated into English by Tomas Diaz. orders@tomasdiaz.com

Solution

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See also

https://www.oma.org.ar/enunciados/ibe18.htm

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