8.5. a) A rectangular table of size $4 \times 10$ (cells) is given. What is the maximum number of crosses that can be placed in the cells of this table so that the following condition is met: in each row and each column of the table, there must be an odd number of crosses? b) Is it possible to place several crosses in a table of size $5 \times 10$ so that the specified condition is met?
Answer: a) 30; b) cannot. Solution. a) Let the table have 4 rows and 10 columns. In each column, there is at least one empty (without a cross) cell, otherwise there would be a column with 4 crosses. Therefore, the total number of crosses in the table is no more than $3 \cdot 10=30$. An example with 30 crosses is shown in the figure. b) Suppose, for the sake of contradiction, that such an arrangement exists for a $5 \times 10$ table. Then the total number of crosses in the table, when counted by rows, is the sum of five odd numbers, i.e., an odd number. On the other hand, when counted by columns, it is the sum of ten odd numbers, i.e., an even number. This is a contradiction.

## 9th grade