Difference between revisions of "2002 OIM Problems/Problem 5"
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== Solution == | == Solution == | ||
− | {{ | + | Notice that every time we apply the recursion, we are essentially subtracting <math>\frac{1}{a_n}</math> from the current term. Clearly, the sequence is decreasing, so for all <math>n</math>, <math>a_n\le56</math>, so <math>\frac{1}{a_n}\ge\frac{1}{56}</math>. Then, after every application of the recursion, the value of <math>a_i</math> will decrease by at least <math>\frac{1}{56}</math>; clearly, it will eventually reach negative numbers. |
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+ | ~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406] | ||
== See also == | == See also == | ||
https://www.oma.org.ar/enunciados/ibe18.htm | https://www.oma.org.ar/enunciados/ibe18.htm |
Latest revision as of 18:23, 5 May 2025
Problem
The sequence of real numbers is defined as:
for every integer .
Prove that there exists an integer ,
, such that
.
~translated into English by Tomas Diaz. orders@tomasdiaz.com
Solution
Notice that every time we apply the recursion, we are essentially subtracting from the current term. Clearly, the sequence is decreasing, so for all
,
, so
. Then, after every application of the recursion, the value of
will decrease by at least
; clearly, it will eventually reach negative numbers.