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.
Step 1 of 5: Square gap
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a2+b+c≥(a+1)2a^2+b+c\ge(a+1)^2
Detailed analysis

If a2+b+ca^2+b+c is a square, it is strictly larger than a2a^2; hence it is at least (a+1)2(a+1)^2, so b+c≥2a+1b+c\ge2a+1.