6
Problems(2)
game with 15 totem poles in an Indian reservatory
Source: 2021 Dürer Math Competition Finals Day 1 E6 E+6
1/2/2022
(Game) In an Indian reservatory there are totem poles arranged according to the left figure. Silent Stream and Red Fire used to play the following game: In turns they stretch ropes between two-two poles in such a way that every stretched rope is parallel to a side of the big triangle and no rope can go along a pole that is already touched by another rope. Furthermore, if instead of a rope one can stretch out a straight line extension of the rope, then one should stretch out this extension. The one who cannot stretch out more rope according to the rules loses.
Win two games in a row against the organizers! You can decide that you want to start or to be the second player. The figure on the right depicts the first three steps of a game. First Silent Stream stretches the blue rope, then Red Fire stretches the red one, finally Silent Stream stretches the blue one. https://cdn.artofproblemsolving.com/attachments/f/8/3b8b9e38a8a477da288566ecb26036bfc7e615.png
combinatoricsgame
4-digit positive number. and simple graph on 4 vertices
Source: 2021 Dürer Math Competition Finals Day2 E6 https://artofproblemsolving.com/community/c2749870_
1/8/2022
Bertalan thought about a -digit positive number. Then he draw a simple graph on vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.
combinatoricsgraph theorygraph