BALKAN-SL 2009 A3

Denote by $S(x)$ the sum of digits of positive integer $x$ written in decimal notation. For $k$ a fixed positive integer, define a sequence $(x_n)_{n \geq 1}$ by $x_1=1$ and $x_{n+1}$ $=$ $S(kx_n)$ for all positive integers $n$ . Prove that $x_n$ $<$ $27 \sqrt{k}$ for all positive integer $n$ .