CZ-SK-MO 2003 Problem 5

Show that, for each integer $z \ge 3$ , there exist two two-digit numbers $A$ and $B$ in base $z$ , one equal to the other one read in reverse order, such that the equation $x^2 -Ax+B$ has one double root. Prove that this pair is unique for a given $z$ . For instance, in base $10$ these numbers are $A = 18, B = 81$ .