Školjka

%V0 In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.

%V0 Let $ABCD$ be a regular tetrahedron and $M,N$ distinct points in the planes $ABC$ and $ADC$ respectively. Show that the segments $MN,BN,MD$ are the sides of a triangle.

%V0 Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $120^{\circ}.$

%V0 Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

%V0 Let $Ox, Oy, Oz$ be three rays, and $G$ a point inside the trihedron $Oxyz$. Consider all planes passing through $G$ and cutting $Ox, Oy, Oz$ at points $A,B,C$, respectively. How is the plane to be placed in order to yield a tetrahedron $OABC$ with minimal perimeter ?

%V0 Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. (a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. (b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the product of the three smallest edges not belonging to the same face.