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ax^2 + by^2 + cz^2 = 1

Source: IMO LongList 1988, USS 3, Problem 86 of ILL

November 9, 2005
number theory solvednumber theory

Problem Statement

Let a,b,ca,b,c be integers different from zero. It is known that the equation a⋅x2+b⋅y2+c⋅z2=0a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 0 has a solution (x,y,z)(x,y,z) in integer numbers different from the solutions x=y=z=0.x = y = z = 0. Prove that the equation a⋅x2+b⋅y2+c⋅z2=1 a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 1 has a solution in rational numbers.
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