5. In each cell of a $15 \times 15$ table, a number $-1, 0$, or $+1$ is written such that the sum of the numbers in any row is non-positive, and the sum of the numbers in any column is non-negative. What is the smallest number of zeros that can be written in the cells of the table?
Answer: 15.

Evaluation. Since the sum of the numbers in all rows is non-positive, the sum of all numbers in the table is also non-positive. On the other hand, if the sum of the numbers in any column is non-negative, then the sum of all numbers in the table is non-negative. Therefore, the sum of all numbers in the table is actually zero, and the sums of the numbers in each column and each row are also zero.

In each row, there must be at least one zero, because otherwise the sum of all fifteen +1 and -1 would be odd, while it should be zero.

Thus, there must be at least 15 zeros in the table.

![](https://cdn.mathpix.com/cropped/2024_05_06_68d12f63c1a28df9188ag-3.jpg?height=579&width=594&top_left_y=1184&top_left_x=731)

Fig. 2: Solution to problem 5.

Example. One possible example with 15 zeros is shown in Fig. 2. The symbols “+” and “-” correspond to the placement of numbers +1 and -1, and the shaded cells represent the placement of zeros.

## Criteria

+ Correct solution - 7 points.

-. Correct answer given without explanations (or with incorrect explanations) - 1 point.

The solution naturally divides into two parts: an example of number placement with 15 zeros and a proof that there are at least 15 zeros. Points for these parts are summed.

干 Correct example with 15 zeros provided - 3 points.

$\pm$ It is justified that there cannot be fewer than 15 zeros (or that there is at least one zero in each row/column) - 4 points.

- It is noted that the sum of the numbers in each row and each column must be zero, but the conclusion about the presence of a zero in any row/column did not follow - 1 point out of 4 possible for the estimate.