3
Part of 2016 Israel Team Selection Test
Problems(2)
Three-Dimensional Brianchon
Source: Israel Autumn 2016 TST1/3
9/29/2016
Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).
geometryconics3D geometryoctahedron
Save the Dragons
Source: Israel Autumn 2016 TST2/3
9/29/2016
On each square of an x board sleeps a dragon. Two dragons are called neighbors if their squares have a side in common. Each turn, Minnie wakes up a dragon which has a living neighbor and Max directs it towards one of its living neighbors. The dragon than breathes fire on that neighbor and destroys it, and then goes back to sleep. Minnie's goal is to minimize the snoring of the dragons and leave as few living dragons as possible. Max is a member of PETD (People for the Ethical Treatment of Dragons), and he wants to save as many dragons as he can.How many dragons will stay alive at the end if1. ?
2. ?
combinatoricscombinatorics solved