PL-OM 2001 Problem 3

A sequence $x_0=A$ and $x_1=B$ and $x_{n+2}=x_{n+1} +x_n$ is called a Fibonacci type sequence. Call a number $C$ a repeated value if $x_t=x_s=c$ for $t$ different from $s$ . Prove one can choose $A$ and $B$ to have as many repeated value as one likes but never infinite.