MathDB
A n x 6 - array

Source: Baltic Way 2005/7

November 7, 2005
vectorfloor functioncombinatorics proposedcombinatoricsBaltic Way2005#7

Problem Statement

A rectangular array has n n rows and 6 6 columns, where n2 n \geq 2. In each cell there is written either 0 0 or 1 1. All rows in the array are different from each other. For each two rows (x1,x2,x3,x4,x5,x6) (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}) and (y1,y2,y3,y4,y5,y6) (y_{1},y_{2},y_{3},y_{4},y_{5},y_{6}), the row (x1y1,x2y2,x3y3,x4y4,x5y5,x6y6) (x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6}) can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.