2023 IMO Problems/Problem 3
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]
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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
: