## 52. Five Numbers

Try to find five consecutive integers such that the sum of the squares of the two largest of them equals the sum of the squares of the other three?
52. Let $n$ be the average of our five numbers; then the other numbers are $(n+1)$ and $(n+2); (n-1)$ and $(n-2)$. According to the problem, the following relationship holds:

$$
(n+1)^{2}+(n+2)^{2}=n^{2}+(n-1)^{2}+(n-2)^{2}
$$

Simplifying this equation, we get

$$
n(n-12)=0
$$

The only acceptable solution is

$$
n=12
$$

Thus, the five numbers we are looking for are: $10,11,12,13,14$.