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r_i +r_j -n is divisible by m

Source: Polish MO Finals 1979 p1

August 24, 2024

number theorydivisible

Problem Statement

Let be given a set

\{r_1,r_2,...,r_k\}

of natural numbers that give distinct remainders when divided by a natural number

m

. Prove that if

k > m/2

, then for every integer

n

there exist indices

i

and

j

(not necessarily distinct) such that

r_i +r_j -n

is divisible by

m