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(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2, (a^2 + d^2)(b^2 + c^2) = (ad + bc)^2

Source: 2012 Cuba MO 2.8

September 18, 2024
number theoryPerfect Square

Problem Statement

If the natural numbers a,b,c,da, b, c, d verify the relationships: (a2+b2)(c2+d2)=(ab+cd)2(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2 (a2+d2)(b2+c2)=(ad+bc)2(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2 and the gcd(a,b,c,d)=1gcd(a, b, c, d) = 1, prove that a+b+c+da + b + c + d is a perfect square.
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