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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?

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If you're only proving it for ten points, why can't you just compute the values of $f(x)$ at each of $x =1, \ldots, 10$? –  Christopher A. Wong Oct 30 '12 at 7:50
Thanks Chris for the reply. Yes, that can be done when $k$ is fixed. I did modify the question to be more general. –  hanna Oct 30 '12 at 13:08

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