## Task 1 - 261211

Determine all triples $(x ; y ; z)$ of numbers with the following properties (1), (2):

(1) The numbers $x, y, z$ are consecutive integers in this order.

(2) It holds: $x \cdot(x+y+z)=x \cdot y \cdot z$.
I. If $(x ; y ; z)$ is a triple with properties (1) and (2), then according to (1)

$$
x=n-1 \quad, \quad y=n \quad, \quad z=n+1
$$

with an integer $n$, so $x+y+z=3 n$, and from (2) it follows

$$
\begin{aligned}
(n-1) \cdot 3 \cdot n & =(n-1) \cdot n \cdot(n+1) \\
n \cdot(n-1) \cdot(n-2) & =0
\end{aligned}
$$

Since a product can only be 0 if one of the factors is 0, it follows: $n=0$ or $n=1$ or $n=2$. Due to (3), the only possible triples are $(-1 ; 0 ; 1),(0 ; 1 ; 2),(1 ; 2 ; 3)$ that satisfy the required properties.

II. These triples have these properties; for they are triples of consecutive integers $x, y, z$ in this order and they satisfy the verification in equation (2).

## Adopted from $[5]$