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x^2+1/x^2+y^2+1/y^2 rational if x+1/x+y+1/y, x^3+1/x^3+y^3+1/y^3 rationals

Source: Polish Math Olympiad 2021 2nd round p1 day 2

May 31, 2021

algebraretional

Problem Statement

There are real numbers

x, y

such that

x \ne 0

y \ne 0

xy + 1 \ne 0

and

x + y \ne 0

. Suppose the numbers

x + \frac{1}{x} + y + \frac{1}{y}

and

x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}

are rational. Prove that then the number

x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}

is also rational.