Problem 10.5. In each cell of a square table of size $200 \times 200$, a real number not exceeding 1 in absolute value was written. It turned out that the sum of all the numbers is zero. For what smallest $S$ can we assert that in some row or some column, the sum of the numbers will definitely not exceed $S$ in absolute value?
Answer: 100.

Solution. First, we show that $S<40000$.
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This means that one of the numbers $A$ or $D$ in absolute value exceeds 10000. However, each of the corresponding squares contains only 10000 cells, and the numbers in them do not exceed 1 in absolute value. Contradiction.

## Criteria

Any correct solution to the problem is worth 7 points. In the absence of such a solution, the following criteria are summed:

## 16. There is a correct answer.

1 6. "Example". It is proven that $S<100$ is impossible (i.e., an example is provided or its existence is proven).

5 6. "Estimate". It is proven that $S=100$ works for any arrangement of numbers satisfying the condition. In the absence of a complete proof of the "estimate", the following partial progress is credited: