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solving equations in integers, quadratics

Source: Spanish Mathematical Olympiad 1992 P3

August 1, 2018
quadraticsnumber theory

Problem Statement

Prove that if a,b,c,da,b,c,d are nonnegative integers satisfying (a+b)2+2a+b=(c+d)2+2c+d(a+b)^2+2a+b= (c+d)^2+2c+d, then a=ca = c and b=db = d. Show that the same is true if a,b,c,da,b,c,d satisfy (a+b)2+3a+b=(c+d)2+3c+d(a+b)^2+3a+b=(c+d)^2+3c+d, but show that there exist a,b,c,da,b,c,d with a≠ca \ne c and b≠db \ne d satisfying (a+b)2+4a+b=(c+d)2+4c+d(a+b)^2+4a+b = (c+d)^2+4c+d.
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