MathDB

2

Part of 2014 European Mathematical Cup

Problems(2)

Jemc p2

Source: European Mathematical Cup 2014, Junior Division, Problem 2

12/22/2014
In each vertex of a regular nn-gon A1A2...AnA_1A_2...A_n there is a unique pawn. In each step it is allowed: 1. to move all pawns one step in the clockwise direction or 2. to swap the pawns at vertices A1A_1 and A2A_2. Prove that by a finite series of such steps it is possible to swap the pawns at vertices: a) AiA_i and Ai+1A_{i+1} for any 1≤i<n 1 \leq i < n while leaving all other pawns in their initial place b) AiA_i and AjA_j for any 1≤i<j≤n 1 \leq i < j \leq n leaving all other pawns in their initial place.
Proposed by Matija Bucic
group theorycombinatorics unsolvedcombinatorics
Chess game with knight and queen

Source: European Mathematical Cup 2014, Senior Division, P2

12/14/2014
Jeck and Lisa are playing a game on table dimensions m×nm \times n , where m,n>2m , n >2. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules: 1. Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table 2. Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table. Player which cannot put his figurine loses. For which pairs of (m,n)(m,n) Lisa has winning strategy?
Proposed by Stijn Cambie
geometryrectanglecombinatorics unsolvedcombinatorics