MathDB

4

Part of 1999 Bundeswettbewerb Mathematik

Problems(2)

min no of triangular faces of polyhedron with faces more numbered than vertices

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p4

1/27/2020
It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.
combinatorial geometrygeometrycombinatorics
bright number is a sum of a perfect square and a perfect cube

Source: 1999 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p4

1/27/2020
A natural number is called bright if it is the sum of a perfect square and a perfect cube. Prove that if rr and ss are any two positive integers, then (a) there exist infinitely many positive integers nn such that both r+nr+n and s+ns+n are bright, (b) there exist infinitely many positive integers mm such that both rm and sm are bright.
number theoryPerfect Squareperfect cube