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Part of 2022 Iran-Taiwan Friendly Math Competition

Problems(1)

aops compiled \lfloor :)

Source: 2022 IRN TWN Friendly math competition P1

k\geqslant 2

be an integer, and

a,b

be real numbers. prove that

a-b

is an integer divisible by

k

if and only if for every positive integer

n

\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)

Proposed by Navid Safaei

number theoryfloor functionIranTaiwan