Cho x + y + z = 0. Rút gọn: \(A = \frac{{{x^2} + {y^2} + {z^2}}}{{{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}}}\).

Ta có: x + y + z = 0

Þ (x + y + z)² = 0

Û x² + y² + z² + 2xy + 2yz + 2zx = 0 (1)

Thay (1) vào A ta được:

\(A = \frac{{{x^2} + {y^2} + {z^2}}}{{{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}}}\)

\( = \frac{{{x^2} + {y^2} + {z^2}}}{{3\left( {{x^2} + {y^2} + {z^2}} \right) - \left( {{x^2} + {y^2} + {z^2} + 2xy + 2yz + 2zx} \right)}}\)

\( = \frac{{{x^2} + {y^2} + {z^2}}}{{3\left( {{x^2} + {y^2} + {z^2}} \right)}} = \frac{1}{3}\)