MathDB

86

Part of 1988 IMO Longlists

Problems(1)

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

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

11/9/2005
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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