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ab, bc, ca, ab + bc + ca perfect squares if a^2+b^2+c^2=(a-b)^2+(b-c)^2+(c-a)^2

Source: 15th -a QEDMO problem 3 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 29, 2021
number theoryPerfect Square

Problem Statement

Let a,b,ca,b,c natural numbers for which a2+b2+c2=(a−b)2+(b−c)2+(c−a)2a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2. Show that ab,bc,caab, bc, ca and ab+bc+caab + bc + ca are perfect squares .
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