AIME 1984

Determine $x^2 + y^2 + z^2 + w^2$ if

$\frac {x^2}{2^2 - 1^2} + \frac {y^2}{2^2 - 3^2} + \frac {z^2}{2^2 - 5^2} + \frac {w^2}{2^2 - 7^2} = 1$

$\frac {x^2}{4^2 - 1^2} + \frac {y^2}{4^2 - 3^2} + \frac {z^2}{4^2 - 5^2} + \frac {w^2}{4^2 - 7^2} = 1$

$\frac {x^2}{6^2 - 1^2} + \frac {y^2}{6^2 - 3^2} + \frac {z^2}{6^2 - 5^2} + \frac {w^2}{6^2 - 7^2} = 1$

$\frac {x^2}{8^2 - 1^2} + \frac {y^2}{8^2 - 3^2} + \frac {z^2}{8^2 - 5^2} + \frac {w^2}{8^2 - 7^2} = 1$