1 Autumn tournament

a) Example: we number the rows and columns from $1$ to $9$. We write $1$ in the fields $(i,i)$ ($i=1,\dots ,9$); $-1$ in field $(1,9)$ and in fields $(i,i-1)$ ($i=2,\dots ,9$); $0$ in other fields. Possible sums are $1$, $0$, and $-1$.

Evaluation: Suppose there are at least $19$ non-zero numbers. From Dirichlet's principle, there will be at least three non-zero numbers on any row, and therefore at least two non-zero numbers with the same sign, let's say positive ones (the situation with negative ones is analogous). Let's arrange the numbers in this order by size: $a_1\le a_2\le \cdots \le a_9$, where $a_9\ge a_8>0$. If $a_5\ge 0$, then $a_5+a_6+a_7+a_8+a_9\ge a_8+a_9>a_9$ must be of the same order, a contradiction. If $a_5<0$, then $a_1+a_2+a_3+a_4+a_5<a_1$ must be of the same order, a contradiction.

b) A possible example without zeros is as follows (works because $5\cdot 3+3\cdot (-4)=3$ and $4\cdot 3+4\cdot (-4)=-4$):

3	3	3	3	3	-4	-4	-4	-4
-4	3	3	3	3	3	-4	-4	-4
-4	-4	3	3	3	3	3	-4	-4
-4	-4	-4	3	3	3	3	3	-4
-4	-4	-4	-4	3	3	3	3	3
3	-4	-4	-4	-4	3	3	3	3
3	3	-4	-4	-4	-4	3	3	3
3	3	3	-4	-4	-4	-4	3	3
3	3	3	3	3	-4	-4	-4	-4