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>f(n)</math> and <math>g(i)</math> be functions of positive integers n and i respectively. | ||
Let <math>a_{n}=a_{1}+f(n)</math>, then <math>a_{n+1}=a_{1}+f(n+1)</math>, <math>a_{n+k}=a_{1}+f(n+k)</math> | Let <math>a_{n}=a_{1}+f(n)</math>, then <math>a_{n+1}=a_{1}+f(n+1)</math>, <math>a_{n+k}=a_{1}+f(n+k)</math> | ||
Let <math>P=\prod_{i=1}^{k}\left ( a_{n+i} \right ) = \prod_{i=1}^{k}\left ( a_{n}+g(i)) \right )</math> | Let <math>P=\prod_{i=1}^{k}\left ( a_{n+i} \right ) = \prod_{i=1}^{k}\left ( a_{n}+g(i)) \right )</math> |
Revision as of 11:30, 3 October 2023
Problem
For each integer , determine all infinite sequences of positive integers for which there exists a polynomial of the form , where are non-negative integers, such that for every integer .
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 and be functions of positive integers n and i respectively.
Let , then ,
Let