1. a) If a and b are two integers, determine the set $A=\left\{a^{b} \mid a b-1=a+b\right\}$.

b) Does there exist $n \in \mathbb{Z}$ such that the numbers $\frac{3 n+1}{4}$ and $\frac{5 n-3}{4}$ are simultaneously integers?

Vasile Chiș, Reșița
1. a) From the given relation we obtain $b=\frac{a+1}{a-1} \in \mathbb{Z}$

we deduce that $a-1 \mid a+1$ and since $a-1 \mid a-1$ we get that $a-1 \mid 2$ $1 p$

From $a-1 \in D_{2} \Rightarrow a-1 \in\{ \pm 1, \pm 2\}$ and we obtain the solutions $(2,3),(0,-1),(3,2),(-1,0) \ldots .1 \mathrm{p}$ The solution $(0,-1)$ does not work. Final result $A=\{1,8,9\}$

b) Suppose the two numbers were simultaneously integers. We obtain:

$$
\left.\frac{3 n+1}{4} \in \mathbb{Z} \Rightarrow 4|3 n+1 \Rightarrow 4| 5(3 n+1) \Rightarrow 4 \right\rvert\, 15 n+5
$$

and $\left.\frac{5 n-3}{4} \in \mathbb{Z} \Rightarrow 4|5 n-3 \Rightarrow 4| 3(5 n-3) \Rightarrow 4 \right\rvert\, 15 n-9$

Therefore: $4 \mid(15 n+5)-(15 n-9)$ and thus $4 \mid 14$ - absurd

$1 p$