3. In a square table with 2015 rows and columns, positive numbers are arranged. The product of the numbers in each row and each column is 1, and the product of the numbers in any 1008 x 1008 square is 2. What number is in the center of the table?

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# Solution

Consider the first 1008 rows of the table. From the additional condition, it follows that if these rows are covered by two squares of size $1008 \times 1008$, then these squares will overlap by one column. Denote the product of the numbers in this column (1008 numbers) by $M$.

Then the product of all the numbers in the first 1008 rows of the table is, on the one hand, 1, and on the other hand, $2^{2} / M$.

Thus, $\quad M=4$.

Now consider the middle column of the table. It is similarly divided into two blocks of 1008 elements, which overlap at the central element of the table (if moving from the top and bottom towards each other). Denote this element $C$.

The product of all the numbers in this column is 1. Therefore, $1=M^{2} / C$.

From which $C=M^{2}=16$.

Answer. The number in the center of the table is 16.