MathDB
Periodic Sequence with Perfect Square Condition

Source: KMO 2022 P3

October 29, 2022
number theoryPerfect SquaresSequenceInteger sequence

Problem Statement

Suppose that the sequence {an}\{a_n\} of positive integers satisfies the following conditions:
[*]For an integer i≥2022i \geq 2022, define aia_i as the smallest positive integer xx such that x+∑k=i−2021i−1akx+\sum_{k=i-2021}^{i-1}a_k is a perfect square. [*]There exists infinitely many positive integers nn such that an=4×2022−3a_n=4\times 2022-3.
Prove that there exists a positive integer NN such that ∑k=nn+2021ak\sum_{k=n}^{n+2021}a_k is constant for every integer n≥Nn \geq N. And determine the value of ∑k=NN+2021ak\sum_{k=N}^{N+2021}a_k.
Back to ProblemsView on AoPS