100 Followers Problem Combinatorics

Discrete Mathematics Level 4

Find the smallest integer \(k\) with the following property:

Given any real numbers \(a_{1}, a_{2} ,\ldots, a_{d}\) such that \(a_{1}+a_{2}+\cdots +a_{d}=100\) and \( 0 \le a_{i} \le 1\) 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.