Problems(2)
Coins on the table form regular polygons
Source: BWM Germany Round 1, 2015 - Problem 1
4/8/2015
Twelve 1-Euro-coins are laid flat on a table, such that their midpoints form a regular -gon. Adjacent coins are tangent to each other.
Prove that it is possible to put another seven such coins into the interior of the ring of the twelve coins.
geometrypolygonbundeswettbewerbGermany
Coloring squares in a rectangle: Who can win?
Source: Bundeswettbewerb Mathematik 2015 - Round 2 - #1
9/4/2015
Let be positive even integers. A rectangle with side lengths and is split into unit squares.
Anja and Bernd take turns and in each turn they color a square that is made of those unit squares. The person that can't color anymore, loses.
Anja starts. Find all pairs , such that she can win for sure.Extension: Solve the problem for positive integers that don't necessarily have to be even.
Note: The extension actually was proposed at first. But since this is a homework competition that goes over three months and some cases were weird, the problem was changed to even integers.
geometryrectanglecombinatorics