Cateva probleme (clasice) cu numere intregi

[Claudiu Mindrila]

\( 1)\ \left|\begin{array}{c}
a,\ b,\ c\in\mathbb{N}\\
\frac{ab+1}{c}\in\mathbb{N},\ \frac{bc+1}{a}\in\mathbb{N},\ \frac{ca+1}{b}\in\mathbb{N}\end{array}\right|\Longrightarrow a=?,\ b=?,\ c=? \)

\( 2)\ \left|\begin{array}{c}
a,b,c\in\mathbb{Q},\ a,b>0\\
a+\frac{1}{b}\in\mathbb{Z},\ b+\frac{1}{c}\in\mathbb{Z},\ c+\frac{1}{a}\in\mathbb{Z}\end{array}\right|\Longrightarrow a=?,\ b=?,\ c=? \)

\( 3)\ \left|\begin{array}{c}
a,\ b,\ c\in\mathbb{Z}\\
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\in\mathbb{Z}\\
\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\in\mathbb{Z}\end{array}\right|\Longrightarrow\left|a\right|=\left|b\right|=\left|c\right| \)

[Marius Mainea]

b) \( (1,1,1) \) , \( (1,\frac{1}{2},2) \) , \( (2,\frac{1}{3},\frac{3}{2}) \)