Minimum and Maximum

Chef has $K$ chocolates and he wants to distribute them to $N$ people (numbered $1$ through $N$). These people are standing in a line in such a way that for each $i$ ($1 \le i \le N-1$), person $i$ and person $i+1$ are adjacent. First, consider some way to distribute chocolates such that for each valid $i$, the number of chocolates the $i$-th person would receive from Chef is $A_i$ and the sum $S