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Sets of integers with i+a in A or i-b in B

Source: APMO 2013, Problem 4

May 3, 2013

algebrapolynomialcombinatoricsAPMO

Problem Statement

Let

a

and

b

be positive integers, and let

A

and

B

be finite sets of integers satisfying (i)

A

and

B

are disjoint; (ii) if an integer

i

belongs to either to

A

or to

B

, then either

i+a

belongs to

A

or

X

belongs to

B

. Prove that

a\left\lvert A \right\rvert = b \left\lvert B \right\rvert

. (Here

\left\lvert X \right\rvert

denotes the number of elements in the set

X

.)

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