Building a Tall Barn - Problem Statement

You're building an $N$-story barn with $K$ cows. Each cow is assigned to exactly one floor, each floor has at least one cow. Floor $i$ requires $a_i$ work, and with $c$ cows it takes $a_i / c$ time. Find the lowest total time. Example: $3$ floors with work [$10$, $20$, $30$], $5$ cows.

If you assign $1$, $2$, $2$ cows to floors $1$, $2$, $3$, total time = $10/1 + 20/2 + 30/2 = 10 + 10 + 15 = 35$. This problem beautifully illustrates why convexity holds. Adding more cows gives diminishing returns. Think about it: the first cow on a floor cuts time by half, the second cuts it by a third, etc.