MathLabs

Problem 1

Let a,b,ca,b,c be positive integers. Prove that it is impossible for all three numbers a2+b+ca^2+b+c, b2+c+ab^2+c+a, and c2+a+bc^2+a+b to be perfect squares.
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2(a+b+c)≥2(a+b+c)+32(a+b+c)\ge2(a+b+c)+3
Detailed analysis

Adding the three inequalities gives 2(a+b+c)≥2(a+b+c)+32(a+b+c)\ge2(a+b+c)+3, impossible.