Olympiad problems

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.

1956 Moscow Mathematical Olympiad, 345

Tags: pyramid , 3d geometry , equal faces , trihedral angle , geometry

* 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.

1989 Romania Team Selection Test, 4

Tags: geometry , fixed , tangent spheres , 3d geometry , trihedral angle , sphere

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.

1952 Moscow Mathematical Olympiad, 220

Tags: geometry , 3d geometry , trihedral angle , plane , perpendicular

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 $OS$.