4

Part of 2016 Bundeswettbewerb Mathematik

Problems(2)

Two children with the same name in one class

Source: Germany 2016 - BWM Round 1, #4

33

children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers

0,1,2,\dots,10

appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

combinatoricscombinatorics unsolvedpigeonhole principlegraph theoryGermanyblackboard

Dodecahedron's Planes cuts Space in Regions

Source: Bundeswettbewerb Mathematik 2016, Round 2 - #4

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint regions. Find the number of such regions.

geometry3D geometrydodecahedroncombinatoricscombinatorial geometry