CN-LEAGUE 2009 Problem 4

Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$ . The following conditions are given:

• Every row consists of distinct numbers;
• $\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$ ;
• $x_{17}=x_{28}=x_{39}=0$ ;
• $x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$ .
• The first three columns of $P$ satisfy the following property $(R)$ : for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$ , $1\le k\le 9$ , there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$ .

Prove that: a) the elements $u_1,u_2,u_3$ come from three different columns; b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$ , where $l\ge 4$ , satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$ , then this column uniquely exists.