Olympiad problems

1993 Abels Math Contest (Norwegian MO), 4

Each of the $8$ vertices of a given cube is given a value $1$ or $-1$. Each of the $6$ faces is given the value of product of its four vertices. Let $A$ be the sum of all the $14$ values. Which are the possible values of $A$?

1999 Poland - Second Round, 2

Tags: tiling , combinatorial geometry , cube , combinatorics , geometry , 3d geometry

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

2004 Nicolae Păun, 3

Tags: geometry , 3d geometry , sphere , cube , locus

[b]a)[/b] Show that the sum of the squares of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [b]b)[/b] Show that the sum of the cubes of the minimum distances from a point that is situated on a sphere to the faces of the cube that circumscribe the sphere doesn't depend on the point. [i]Alexandru Sergiu Alamă[/i]

2003 May Olympiad, 5

Tags: 3d geometry , cube , minimum , geometry

An ant, which is on an edge of a cube of side $8$, must travel on the surface and return to the starting point. It's path must contain interior points of the six faces of the cube and should visit only once each face of the cube. Find the length of the path that the ant can carry out and justify why it is the shortest path.

2008 Princeton University Math Competition, B4

Tags: cube , sphere

A cube is divided into $27$ unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these $8$ spheres. What is the smallest possible value for the radius of the last sphere?