Školjka

%V0 Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. let $V_1$ be the volume of the cone and $V_2$ be the volume of the cylinder. a) Prove that $V_1 \neq V_2$; b) Find the smallest number $k$ for which $V_1=kV_2$; for this case, construct the angle subtended by a diamter of the base of the cone at the vertex of the cone.

%V0 The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.

%V0 Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

%V0 Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.

%V0 Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0.$$ Determine when equality occurs.

%V0 Prove that there exists a convex 1990-gon with the following two properties : a.) All angles are equal. b.) The lengths of the 1990 sides are the numbers $1^2$, $2^2$, $3^2$, $\cdots$, $1990^2$ in some order.