Difference between revisions of "2023 IMO Problems/Problem 3"
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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> | ||
| − | If we want the coefficients of <math>P(a_{n})</math> to be positive, then <math>g(i)\geq 0</math> for all <math>i</math> | + | If we want the coefficients of <math>P(a_{n})</math> to be positive, then <math>g(i)\geq 0</math> for all <math>i</math> which will give the following value for <math>P</math>: |
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| + | <math>P=a_{n}^{k}+C_{k-1}a_{n}^{k-1}+...+C_{1}a_{n}+\prod_{i=1}^{k} g(i)</math> | ||
Revision as of 11:40, 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
![\[P\left(a_n\right)=a_{n+1} a_{n+2} \cdots a_{n+k}\]](http://latex.artofproblemsolving.com/8/6/e/86e9095fe41091db78cfab1b7c22211df21037aa.png)
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 
If we want the coefficients of 
to be positive, then 
for all 
which will give the following value for :
