1. In a $7 \times 7$ table (see figure), the elements of each row and each column form arithmetic progressions. What is the number $x$ in the central cell?

![](https://cdn.mathpix.com/cropped/2024_05_06_618a59e79d203bcbb6eag-1.jpg?height=406&width=483&top_left_y=1573&top_left_x=792)
Answer: 111.

Solution. The middle term of an arithmetic progression is equal to the half-sum of the extreme terms, so in the middle cell of the first row, the number is $(3+143) / 2=73$, and in the middle cell of the last row, the number is $(82+216) / 2=149$.

![](https://cdn.mathpix.com/cropped/2024_05_06_618a59e79d203bcbb6eag-1.jpg?height=405&width=516&top_left_y=2193&top_left_x=770)

The elements of the middle column also form an arithmetic progression, so $x=(73+$ $149) / 2=111$.

Comment. The correct answer obtained by sequentially filling in the table - 7 points. For arithmetic errors with correct reasoning, deduct $2-3$ points.