Numbers on a Table
Source: Spring 2006 Tournament of Towns Junior A-Level #7
April 15, 2015
Problem Statement
Anna and Boris have the same copy of table filled with distinct numbers. After choosing the maximal number in the table, Anna erases the row and the column that contain this number. Then she continue the same operations with a smaller table till it is possible.Boris basically does the same; however, each time choosing the minimal number in a table. Can it happen that the total sum of the numbers chosen by Boris
a) is greater than the total sum of the numbers chosen by Anna? (6 points)
b) is greater than the total sum of any numbers of initial table given that no two of the numbers are in the same row or in the same column? (2 points)