Solve $\frac{1}{x-1}+ \frac{2}{x-2}+ \frac{3}{x-3}+\cdots+\frac{10}{x-10}\geq\frac{1}{2} $ I would appreciate if somebody could help me with the following problem: 
Q: find $x$
$$\frac{1}{x-1}+ \frac{2}{x-2}+ \frac{3}{x-3}+\cdots+\frac{10}{x-10}\geq\frac{1}{2} $$

Let $x$ be any number such that $10 \lt x \lt 21$. Then for each $k\in [1,10]$ we have $21k \gt x \gt k$, hence $20k \gt x-k \gt 0$, so $\frac{k}{x-k} \gt \frac{1}{20}$. Summing from $k=1$ to $10$, we obtain the desired inequality.
