Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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...")
 
Line 4: Line 4:
  
 
(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>.)
 +
 +
==Video Solution==
 +
https://youtu.be/ASV1dZCuWGs (in full gory detail!)

Revision as of 03:59, 18 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 gory detail!)

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_IMO_Problems/Problem_3&oldid=222943"