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Prove that x,y exist

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September 1, 2010

number theoryrelatively primeDivisibilityIMO ShortlistIMO Longlist

Problem Statement

Let

m

be a positive integer and

x_0, y_0

integers such that

x_0, y_0

are relatively prime,

y_0

divides

x_0^2+m

, and

x_0

divides

y_0^2+m

. Prove that there exist positive integers

x

and

y

such that

x

and

y

are relatively prime,

y

divides

x^2 + m

,

x

divides

y^2 + m

, and

x + y \leq m+ 1.

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