Let = minimum total time using cows. Claim: is convex, meaning each additional cow helps less than the previous one. Intuition: with few cows, adding one makes a big difference. One cow per floor means total time equals . Adding a second cow to the busiest floor cuts that floor's time in half. Huge improvement.
With many cows, they're already well-distributed. Adding one more barely changes anything. This diminishing returns property, , is exactly convexity (each additional unit helps less than the previous). The Aliens Trick applies.