A geometric inequality?

Algebra Level 4

${\frac{x^2}{(x+y)(x+z)}+\frac{y^2}{(y+x)(y+z)}+\frac{z^2}{(z+x)(z+y)}<A}$

The above inequality is true for any positive numbers $x, y,$ and $z$ . Find the smallest number $A,$ and prove that your answer is right.