Problem 4. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence of positive real numbers such that $a_{1}=1$ and

$$
\frac{1}{a_{1}+a_{2}}+\frac{1}{a_{2}+a_{3}}+\ldots+\frac{1}{a_{n-1}+a_{n}}=a_{n}-1, \text { for all } n \geq 2
$$

Determine the integer part of the number $A_{n}=a_{n} a_{n+1}+a_{n} a_{n+2}+a_{n+1} a_{n+2}$, where $n \in \mathbb{N}^{*}$.

Nelu Gerg, Slatina