2013 IMO Shortlist C1

Let $n$ be a positive integer. Find the smallest integer $k$ with the following property: Given any real numbers $a_{1}, \ldots, a_{d}$ such that $a_{1}+a_{2}+\cdots+a_{d}=n$ and $0 \leq a_{i} \leq 1$ for $i=1,2, \ldots, d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most 1 . (Poland)

令 $n$ 为正整数。找到具有以下属性的最小整数 $k$：给定任何实数 $a_{1}, \ldots, a_{d}$ 使得 $a_{1}+a_{2}+\cdots+a_{d}=n$ 和 $0 \leq a_{i} \leq 1$ 对于 $i=1,2, \ldots, d$，可以将这些数字划分为 $k$ 组(其中一些可能为空)，这样每组中的数字之和最多为 1 。 (波兰)