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Modulo Sequence of 2021

Source: KJMO 2021 P2

November 13, 2021

modular arithmeticSequenceInteger sequencenumber theory

Problem Statement

Let

\{a_n\}

be a sequence of integers satisfying the following conditions.
[*]

a_1=2021^{2021}

[*]

0 \le a_k < k

for all integers

k \ge 2

[*]

a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k

is multiple of

k

for all positive integers

k

.
Determine the

2021^{2022}

th term of the sequence

\{a_n\}