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divisible by p^n-1 but not p^n

Source: Balkan MO Shortlist N5

September 14, 2021

number theory

Problem Statement

Consider an integer

n\geq 2

and an odd prime

p

. Let

U

be the set of all positive integers

(

strictly

)

less than

p^n

that are not divisible by

p

, and let

N

be the number of elements of

U

. Does there exist permutation

a_1,a_2,\cdots a_N

of the numbers in

U

such that the sum

\sum_{k=1}^N a_ka_{k+1}

,where

a_{N+1}=a_1

, be divisible by

p^{n-1}

but not by

p^n

?

Alexander \ Ivanov \, Bulgaria

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