Školjka

%V0 $(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the centers of these hyperbolas.

%V0 $(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

%V0 Given triangle $ABC$ with points $M$ and $N$ are in the sides $AB$ and $AC$ respectively. If $\dfrac{BM}{MA} +\dfrac{CN}{NA} = 1$ , then prove that the centroid of $ABC$ lies on $MN$ .

%V0 $(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

%V0 Given a triangle $ABC$. Let $G$, $I$, $H$ be the centroid, the incenter and the orthocenter of triangle $ABC$, respectively. Prove that $\angle GIH > 90^{\circ}$.

%V0 Let $ABC$ be a triangle, and let the angle bisectors of its angles $CAB$ and $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $F$, respectively. The lines $AD$ and $BF$ meet the line through the point $C$ parallel to $AB$ at the points $E$ and $G$ respectively, and we have $FG = DE$. Prove that $CA = CB$. Original formulation: Let $ABC$ be a triangle and $L$ the line through $C$ parallel to the side $AB.$ Let the internal bisector of the angle at $A$ meet the side $BC$ at $D$ and the line $L$ at $E$ and let the internal bisector of the angle at $B$ meet the side $AC$ at $F$ and the line $L$ at $G.$ If $GF = DE,$ prove that $AC = BC.$