Školjka

%V0 Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers $$\dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3}$$ at least one is $\leq 2$ and at least one is $\geq 2$

%V0 Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if $$b \tan{\dfrac{w}{2}} \leq c <b$$ In what case does the equality hold?

%V0 Consider a plane $\epsilon$ and three non-collinear points $A,B,C$ on the same side of $\epsilon$; suppose the plane determined by these three points is not parallel to $\epsilon$. In plane $\epsilon$ take three arbitrary points $A',B',C'$. Let $L,M,N$ be the midpoints of segments $AA', BB', CC'$; Let $G$ be the centroid of the triangle $LMN$. (We will not consider positions of the points $A', B', C'$ such that the points $L,M,N$ do not form a triangle.) What is the locus of point $G$ as $A', B', C'$ range independently over the plane $\epsilon$?

%V0 Consider $\triangle OAB$ with acute angle $AOB$. Thorugh a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range over a) the side $AB$; b) the interior of $\triangle OAB$.

%V0 Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if $$a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta})$$ the triangle is isosceles.

%V0 Let $ABC$ be a triangle, and let $P$, $Q$, $R$ be three points in the interiors of the sides $BC$, $CA$, $AB$ of this triangle. Prove that the area of at least one of the three triangles $AQR$, $BRP$, $CPQ$ is less than or equal to one quarter of the area of triangle $ABC$. Alternative formulation: Let $ABC$ be a triangle, and let $P$, $Q$, $R$ be three points on the segments $BC$, $CA$, $AB$, respectively. Prove that $\min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$, where the abbreviation $\left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $P_1P_2P_3$.