1. Solve, in the set of complex numbers, the system of equations $z^{19} w^{25}=1$, $z^{5} w^{7}=1, z^{4}+w^{4}=2$.
Solution. From $z^{5} w^{7}=1$ we get $z^{15} w^{21}=1$, so $\frac{z^{19} w^{25}}{z^{15} w^{21}}=1$, i.e. $z^{4} w^{4}=1$. From the last and the third equation of the system, we get that $z^{4}$ and $w^{4}$ are solutions to the equation $t^{2}-2 t+1=0$, i.e. $z^{4}=w^{4}=1$. According to this, $1=z^{5} w^{7}=z w^{3}$. Further, $z w^{4}=w$ so $z=w$. The solutions are $(1,1),(-1,-1),(i, i),(-i,-i)$.