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x + y + z is integer if x^2 = y + 2 and y^2 = z + 2 and z^2 = x + 2

Source: 2023 NZMO - New Zealand Maths Olympiad Round 2 p5

September 2, 2023

algebrasystem of equations

Problem Statement

Let

x, y

and

z

be real numbers such that:

x^2 = y + 2

, and

y^2 = z + 2

, and

z^2 = x + 2

. Prove that

x + y + z

is an integer.