## - Problem 1

A pucelana sequence is a strictly increasing sequence of sixteen consecutive positive odd numbers, whose sum is a perfect cube. How many pucelana sequences consist only of three-digit numbers?
## Solution:

Let the sequence be $n, n+2, \ldots, n+30$. Then the sum is $\frac{1}{2} \cdot 16(2 n+30)=8(2 n+30)$. Therefore, it is necessary that $2 n+30$ be a perfect cube. Now we need to count the number of such $n$ that are odd and satisfy $101 \leq n \leq 969$. The even cubes between 232 and 1968 are 512, 1000, and 1728, which correspond to values of $n$ of 241, 485, and 849. Therefore, there are exactly three such sequences.