1. Determine $n \in \mathbb{N} *$ such that $\sqrt{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2 n) + 8} \in \mathbb{N}$.

Narcis Gabriel Turcu, Brăila
## Solution.

![](https://cdn.mathpix.com/cropped/2024_06_07_fb875762c504876c3fd8g-2.jpg?height=518&width=1567&top_left_y=1024&top_left_x=229)
For $n=2 \Rightarrow \sqrt{16}=4 \in \mathbb{N}$ ..... $1 p$
For $n=3 \Rightarrow \sqrt{56} \notin \mathbb{N}$ ..... $1 p$
For $n=4 \Rightarrow \sqrt{392} \notin \mathbb{N}$ ..... $1 p$
For $n \geq 5 \Rightarrow U(2 \cdot 4 \cdot 6 \cdot \ldots \cdot(2 n)+8)=8 \Rightarrow \sqrt{2 \cdot 4 \cdot 6 \cdot \ldots \cdot(2 n)+8} \notin \mathbb{N}$ ..... $3 p$