Let $x_1,x_2,\ldots,x_n > 0$ be such that $x_1+x_2+\cdots+x_n < 1$. Is it true that there must exist a positive integer $k$ such that $$\{x_1k\}+\{x_2k\}+\cdots+\{x_nk\} > n-1?$$

This looks closely related to the density of the fractional part. The case $n=1$ is obvious, while $n=2$ appeared in this question.

Note that if we allow $x_1+x_2+\cdots+x_n=1$, the statement no longer holds, e.g. by taking $x_1=x_2=\cdots=1/n$.

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.