ELMO 2003 Problem 4

Let $x,y,z \ge 1$ be real numbers such that $$\frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1.$$ Prove that $$\frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1.$$