TITLE: High sum of fractional parts
QUESTION [6 upvotes]: Let $n\geq 2$ and $x_1,\ldots,x_n > 0$ be such that $x_1+\cdots+x_n =1$. Is it true that there must exist a positive integer $k$ such that $$\{x_1k\}+\cdots+\{x_nk\} = n-1?$$
This looks closely related to the density of the fractional part. 
Note that the quantity $\{x_1k\}+\cdots+\{x_nk\}$ is always an integer, since it equals $k-\lfloor x_1k\rfloor - \dots - \lfloor x_nk\rfloor$. Also, as each term is strictly less than one, $n-1$ is the highest value the sum can take.

REPLY [10 votes]: If the $x_i$ are rational, take the lcm of the denominators  and decrease it by $1$.
If they are not necessarily rational, act similarly: using, e.g., Kronecker's theorem, take a $k$ such that all the $kx_i$ are sufficiently close to integers, and decrease that $k$ by $1$.