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Found problems: 2

2003 Romania National Olympiad, 4

[b]a)[/b] Prove that the sum of all the elements of a finite union of sets of elements of finite cyclic subgroups of the group of complex numbers, is an integer number. [b]b)[/b] Show that there are finite union of sets of elements of finite cyclic subgroups of the group of complex numbers such that the sum of all its elements is equal to any given integer. [i]Paltin Ionescu[/i]

1985 Traian Lălescu, 2.3

Let $ 0\neq\varrho\in\text{Hom}\left( \mathbb{Z}_4,\mathbb{Z}_2\right) ,$ $ \text{id}\neq\iota\in\text{Aut}\left( \mathbb{Z}_4\right) ,$ $ G:=\left\{ (x,y)\in\mathbb{Z}_4^2\big|x-y\in\ker\varrho\right\} , $ and $ \rho_1,\rho_2, $ the canonic projections of $ G $ into $ \mathbb{Z}_4. $ Prove that there exists an unique $ \nu\in\text{Hom}\left( \mathbb{Z}_4,G\right) $ such that $ \rho_1\circ\nu=\text{id} $ and $ \rho_2\circ\nu =\iota . $ Determine numerically this morphism.