MathDB

3

Part of 2006 IberoAmerican

Problems(2)

Problem 3

Source: 21-st Iberoamerican Mathematical Olympiad

5/6/2007
The numbers 1, 2, … ,n21,\, 2,\, \ldots\, , n^{2} are written in the squares of an n×nn \times n board in some order. Initially there is a token on the square labelled with n2.n^{2}. In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number 11 along a path with the minimum number of steps. Then it is moved to the square labelled with 2,2, then to square 3,3, etc, always taking the shortest path, until it returns to the initial square. If the total trip takes NN steps, find the smallest and greatest possible values of N.N.
analytic geometrycombinatorics unsolvedcombinatorics
Problem 6

Source: 21-st Iberoamerican Mathematical Olympiad

5/6/2007
Consider a regular nn-gon with nn odd. Given two adjacent vertices A1A_{1} and A2,A_{2}, define the sequence (Ak)(A_{k}) of vertices of the nn-gon as follows: For k≥3, Akk\ge 3,\, A_{k} is the vertex lying on the perpendicular bisector of Ak−2Ak−1.A_{k-2}A_{k-1}. Find all nn for which each vertex of the nn-gon occurs in this sequence.
functionalgebradomaininductiongeometryperpendicular bisectorstrong induction