MathDB

There are

n!

empty baskets in a row, labelled

1, 2, . . . , n!

. Caesar first puts a stone in every basket. Caesar then puts 2 stones in every second basket. Caesar continues similarly until he has put

n

stones into every nth basket. In other words, for each

i = 1, 2, . . . , n,

Caesar puts

i

stones into the baskets labelled

i, 2i, 3i, . . . , n!.

Let

x_i

be the number of stones in basket

i

after all these steps. Show that

n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i}

2