1. Solve in integers the equation $\sqrt{x-5}=15-y^{2}$.

Daniela Tilincă and Adriana Mihăilă, Brăila
Solution. $\sqrt{x-5} \geq 0$ and $x-5 \geq 0 \Rightarrow 15-y^{2} \geq 0 \Rightarrow y^{2} \leq 15$ and $y \in Z \Rightarrow y^{2} \in\{0,1,4,9\}$

- $y^{2}=0 \Rightarrow \sqrt{x-5}=15 \Rightarrow x-5=225 \Rightarrow x=230$
- $y^{2}=1 ; y= \pm 1 \Rightarrow \sqrt{x-5}=14 \Rightarrow x-5=196 \Rightarrow x=201$
- $y^{2}=4 ; y= \pm 2 \Rightarrow \sqrt{x-5}=11 \Rightarrow x-5=121 \Rightarrow x=126$
- $y^{2}=9 ; y= \pm 3 \Rightarrow \sqrt{x-5}=6 \Rightarrow x-5=36 \Rightarrow x=41$