Question 1
Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: for all real numbers $a_1,\ldots,a_d$ satisfying $a_1+\cdots+a_d=n$ and $0\leq a_i\leq1$, it is possible to group the numbers into $k$ packets (possibly empty) so that every packet has sum at most 1.