Let $k_1+k_2=k$, where $k, k_1$ are all positive integers with $k_1 \ge 1$. Also let $K=\min\{k_1, \lfloor k_2/9 \rfloor +1\}$.
Define $g(x)=\max\{1 \le i \le k: \lfloor i/9 \rfloor +1=x\}, x=1, \ldots, K$. Define for $x=1, \ldots, K$ that \begin{align*} f(x)=\dfrac{\lfloor 0.1(x+g(x)) \rfloor +1}{k-x+\lfloor 0.1(x+g(x)) \rfloor +1} \end{align*}
I would like to show that $f(x)/x$ is nondecreasing for $x=1,\ldots, K$. Can anyone give some hint?