Olympiad problems

1952 Moscow Mathematical Olympiad, 220

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

perpendicular geometry 3d geometry plane trihedral angle solid geometry

1956 Moscow Mathematical Olympiad, 345

* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.

pyramid equal faces geometry 3d geometry trihedral angle solid geometry

1955 Moscow Mathematical Olympiad, 301

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

geometry 3d geometry angle acute trihedral angle solid geometry

1989 Romania Team Selection Test, 4

Let $A,B,C$ be variable points on edges $OX,OY,OZ$ of a trihedral angle $OXYZ$, respectively. Let $OA = a, OB = b, OC = c$ and $R$ be the radius of the circumsphere $S$ of $OABC$. Prove that if points $A,B,C$ vary so that $a+b+c = R+l$, then the sphere $S$ remains tangent to a fixed sphere.

fixed geometry 3d geometry trihedral angle tangent spheres sphere solid geometry