MathDB
I can obtain always integer number of points

Source: Own, from Oliforum math contest, problem 5

September 30, 2009
inductionmodular arithmeticlinear algebramatrixvectoralgebrasystem of equations

Problem Statement

Let X: \equal{} \{x_1,x_2,\ldots,x_{29}\} be a set of 29 29 boys: they play with each other in a tournament of Pro Evolution Soccer 2009, in respect of the following rules: i) every boy play one and only one time against each other boy (so we can assume that every match has the form (xi Vs xj) (x_i \text{ Vs } x_j) for some i≠j i \neq j); ii) if the match (xi Vs xj) (x_i \text{ Vs } x_j), with i≠j i \neq j, ends with the win of the boy xi x_i, then xi x_i gains 1 1 point, and xj x_j doesn’t gain any point; iii) if the match (xi Vs xj) (x_i \text{ Vs } x_j), with i≠j i \neq j, ends with the parity of the two boys, then 12 \frac {1}{2} point is assigned to both boys. (We assume for simplicity that in the imaginary match (xi Vs xi) (x_i \text{ Vs } x_i) the boy xi x_i doesn’t gain any point). Show that for some positive integer k≤29 k \le 29 there exist a set of boys {xt1,xt2,…,xtk}⊆X \{x_{t_1},x_{t_2},\ldots,x_{t_k}\} \subseteq X such that, for all choice of the positive integer i≤29 i \le 29, the boy xi x_i gains always a integer number of points in the total of the matches {(xi Vs xt1),(xi Vs xt2),…,(xi Vs xtk)} \{(x_i \text{ Vs } x_{t_1}),(x_i \text{ Vs } x_{t_2}),\ldots, (x_i \text{ Vs } x_{t_k})\}. (Paolo Leonetti)
Back to ProblemsView on AoPS