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Difference between revisions of "2023 IMO Problems/Problem 3"

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https://www.youtube.com/watch?v=CmJn5FKxpPY [Video contains another solution to problem 3]
 
https://www.youtube.com/watch?v=CmJn5FKxpPY [Video contains another solution to problem 3]
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Let <math>a_{n}=a_{1}+f(n)</math>
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<math>a_{n+1}=a_{1}+f(n+1)</math>
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<math>a_{n+k}=a_{1}+f(n+k)</math>

Revision as of 11:25, 3 October 2023

Problem

For each integer $k \geqslant 2$, determine all infinite sequences of positive integers $a_1, a_2, \ldots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1} x^{k-1}+\cdots+c_1 x+c_0$, where $c_0, c_1, \ldots, c_{k-1}$ are non-negative integers, such that \[P\left(a_n\right)=a_{n+1} a_{n+2} \cdots a_{n+k}\] for every integer $n \geqslant 1$.

Solution

https://www.youtube.com/watch?v=JhThDz0H7cI [Video contains solutions to all day 1 problems]

https://www.youtube.com/watch?v=SP-7LgQh0uY [Video contains solution to problem 3]

https://www.youtube.com/watch?v=CmJn5FKxpPY [Video contains another solution to problem 3]

Let $a_{n}=a_{1}+f(n)$

$a_{n+1}=a_{1}+f(n+1)$

$a_{n+k}=a_{1}+f(n+k)$

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