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three interesting numbers

Source: Saint Petersburg MO 2020 Grade 9 Problem 2

May 7, 2020

number theory

Problem Statement

For the triple

(a,b,c)

of positive integers we say it is interesting if

c^2+1\mid (a^2+1)(b^2+1)

but none of the

a^2+1, b^2+1

are divisible by

c^2+1

. Let

(a,b,c)

be an interesting triple, prove that there are positive integers

u,v

such that

(u,v,c)

is interesting and

uv<c^3