There is number in every cell of the table written under such conditions: not all number should be different, however, there is no row or column where all five numbers are equal; the middle number (the third one) in every row and column equals the mean of the numbers in its row or column. What is the minimum amount of numbers that are less than the number written in the middle of the table?
Solution
Let us prove that there cannot be less than three such numbers. Let the middle number of the table be . Then there exists at least one number that is less than in the third row and column. If there is no such number then all the numbers in the row or column are the same, which contradicts the conditions. Without loss of generality let us consider that the number is situated in the middle row in the fifth column. But then in the fifth column exists the number smaller than , and smaller than . Therefore, at least three such numbers exist.
The example where answer 3 is achievable is shown on the picture (fig. 17).
| 4 | 4 | 3 | 4 | 0 |
|---|---|---|---|---|
| 4 | 4 | 3 | 4 | 0 |
| 3 | 3 | 0 | 3 | -9 |
| 4 | 4 | 3 | 4 | 0 |
| 0 | 0 | -9 | 0 | -36 |
Fig. 17
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