Difference between revisions of "2023 IMO Problems/Problem 3"
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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) = P(a_{n})</math> | <math>P=a_{n}^{k}+C_{k-1}a_{n}^{k-1}+...+C_{1}a_{n}+\prod_{i=1}^{k} g(i) = P(a_{n})</math> | ||
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+ | Thus for every <math>i</math> and <math>n</math> we need the following: | ||
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+ | <math>a_{n}+g(i)=a_{n+i}=a_{1}+f(n+i)</math> |
Revision as of 12:45, 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
If we want the coefficients of to be positive, then
for all
which will give the following value for
:
Thus for every and
we need the following: