Problem 8.6. In a $3 \times 3$ table, real numbers are arranged. It turns out that the product of the numbers in any row and any column is 10, and the product of the numbers in any $2 \times 2$ square is 3. Find the number in the central cell.
Answer: $0.00081$.

## Solution.

Let's denote the numbers in the square from left to right: let $a, b, c$ be the numbers in the first row, $d, e, f$ in the second row, and $g, h, i$ in the third row. Notice that

$$
e=\frac{a b d e \cdot b c e f \cdot d e g h \cdot e f h i}{a b c \cdot \operatorname{def} \cdot g h i \cdot b e h \cdot d e f}=\frac{3^{4}}{10^{5}}=0.00081
$$