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 $x$
$$\frac{1}{x-1}+ \frac{2}{x-2}+ \frac{3}{x-3}+\cdots+\frac{10}{x-10}\geq\frac{1}{2} $$

If the left side is $$f(x)=\sum_{k=1}^{10} \frac{k}{x-k},$$
then the graph of $f$ shows that $f(x)<0$ on $(-\infty,1)$, so no solutions there. For each $k=1..9$ there is a vertical asymptote at $x=k$ with the value of $f(x)$ coming down from $+\infty$ immediately to the right of $x=k$ and crossing the line $y=1/2$ between $k$ and $k+1$, afterwards remaining less than $1/2$ in the interval $(k,k+1)$. 
This gives nine intervals of the form $(k,k+a_k]$ where $f(x) \ge 1/2$, and there is a tenth interval in which $f(x) \ge 1/2$ beginning at $x=10$ of the form $(10,a_{10}]$ where $a_{10}$ lies somewhere in the interval $[117.0538,117.0539].$ The values of the $a_k$ for $k=1..9$ are all less than $1$, starting out small and increasing with $k$, some approximations being
$$a_1=0.078,\ a_2=0.143,\ a_3=0.201,\ ...\ a_9=0.615.$$
The formula for finding the exact values of the $a_k$ for $k=1..10$ is a tenth degree polynomial equation which maple12 could not solve exactly, hence the numerical solutions above.
