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A strange divisibility condition

Source: 2024 Nepal TST P3

April 12, 2024

number theory

Problem Statement

Prove that there are infinitely many integers

k\geqslant 2024

for which there exists a set

\{a_1,\ldots,a_k\}

with the following properties: [*]

a_1{}

is a positive integer and

a_{i+1}=a_i+1

for all

1\leqslant i<k,

and [*]

2(a_1\cdots a_{k-2}-1)^2

is divisible by

2(a_1+\cdots+a_k)+a_1-a_1^2.

(Proposed by Prajit Adhikari, Nepal)

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