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Sum of elements of sets give equal residue modulo a prime

Source: 2022 Bulgarian Spring Math Competition, Problem 12.4

March 27, 2022

number theoryprime numbersabstract algebra

Problem Statement

Let

m

and

n

be positive integers and

p

be a prime number. Find the greatest positive integer

s

(as a function of

m,n

and

p

) such that from a random set of

mnp

positive integers we can choose

snp

numbers, such that they can be partitioned into

s

sets of

np

numbers, such that the sum of the numbers in every group gives the same remainder when divided by

p