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c^2/(a + b),b^2/(a + c),a^2/(c + b) integers, a + c, b + c, a + b coprime

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 Shortlist N4 day2

September 30, 2021

number theoryIntegerscoprime

Problem Statement

Show that there are no triples

(a, b, c)

of positive integers such that a)

a + c, b + c, a + b

do not have common multiples in pairs. b)

\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}

are integer numbers.