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Difference between revisions of "2024 IMO Problems/Problem 3"

(Created page with "Let <math>a_1, a_2, a_3, \dots</math> be an infinite sequence of positive integers, and let <math>N</math> be a positive integer. Suppose that, for each <math>n > N</math>, <m...")
 
 
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(An infinite sequence <math>b_1, b_2, b_3, \dots</math> is eventually periodic if there exist positive integers <math>p</math> and <math>M</math> such that <math>b_{m+p} = b_m</math> for all <math>m \ge M</math>.)
 
(An infinite sequence <math>b_1, b_2, b_3, \dots</math> is eventually periodic if there exist positive integers <math>p</math> and <math>M</math> such that <math>b_{m+p} = b_m</math> for all <math>m \ge M</math>.)
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==Video Solution==
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https://youtu.be/ASV1dZCuWGs (in full detail!)
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==See Also==
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{{IMO box|year=2024|num-b=2|num-a=4}}

Latest revision as of 12:24, 30 July 2024

Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$.

Prove that at least one of the sequence $a_1, a_3, a_5, \dots$ and $a_2, a_4, a_6, \dots$ is eventually periodic.

(An infinite sequence $b_1, b_2, b_3, \dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \ge M$.)

Video Solution

https://youtu.be/ASV1dZCuWGs (in full detail!)

See Also

2024 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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