1. (7p) Consider the sets $A=\{n \in N \mid \sqrt{50-5 \sqrt{n+5}} \in N\}$ and $B=\left\{\overline{a b} \left\lvert\, \sqrt{\overline{a b}-a b}=\frac{n}{5}\right., n \in A\right\}$. Determine the sets $A$ and $B$.

Simona Dumitrescu
1. $50-5 \sqrt{n+5}$ is a perfect square and a multiple of 5, so $50-5 \sqrt{n+5} \in\{0,25\}$ (1p)

It follows that $n \in\{20,95\}$, hence

$A=\{20,95\}$ (2p)

$n=95 \Rightarrow \overline{a b}-a b=361$, which is false $\qquad$
(0.5p)

$n=20 \Rightarrow \overline{a b}-a b=16$ and since $b \neq 10 \Rightarrow a=\frac{16-b}{10-b}, a \in \mathbb{N}$

(1.5p)

$b \in\{4,7,8,9\}$,

thus

$B=\{24,37,48,79\}$

(2p)