(HUN) Solve the following system of equations:
$$\begin{cases}x+y+z=a,\\x^2+y^2+z^2=b^2,\\xy=z^2,\end{cases}$$
where $a$ and $b$ are given real numbers. What conditions must hold on $a$ and $b$ for the solutions to be positive and distinct?
(GDR) In the interior of $\triangle P_1P_2P_3$ a point $P$ is given. Let $Q_1,Q_2$, and $Q_3$ respectively be the intersections of $PP_1,PP_2$, and $PP_3$ with the opposing edges of $\triangle P_1P_2P_3$. Prove that among the ratios $PP_1/PQ_1,PP_2/PQ_2$, and $PP_3/PQ_3$ there exists at least one not larger than $2$ and at least one not smaller than $2$.
(CZS) Construct a triangle $ABC$ if the following elements are given: $AC=b$, $AB=c$, and $\angle AMB=\omega$ ($\omega<90^\circ$), where $M$ is the midpoint of $BC$. Prove that the construction has a solution if and only if
$$b\tan\frac\omega2\le c<b.$$
In what case does equality hold?
(ROM) A plane $\epsilon$ is given and on one side of the plane three noncollinear points $A,B$, and $C$ such that the plane determined by them is not parallel to $\epsilon$. Three arbitrary points $A',B'$, and $C'$ in $\epsilon$ are selected. Let $L,M$, and $N$ be the midpoints of $AA',BB'$, and $CC'$, and $G$ the centroid of $\triangle LMN$. Find the locus of all points obtained for $G$ as $A',B'$, and $C'$ are varied (independently of each other) across $\epsilon$.