IMO 1971 LL NET30

Prove that the system of equations

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IMO 1971 LL NET30

Origin: NET

Problem

Prove that the system of equations 2yz + x −y −z = a, 2xz −x + y −z = a, 2xy −x −y + z = a, a being a parameter, cannot have five distinct solutions. For what values of a does this system have four distinct integer solutions?