Problem 8.4. In the cells of a $2 \times 35$ table (2 rows, 35 columns), non-zero real numbers are placed, and all numbers in the top row are distinct. For any two numbers in the same column, the following condition is satisfied: one number is the square of the other.

(a) (1 point) What is the maximum number of negative numbers that can be in this table?

(b) (3 points) What is the minimum number of distinct numbers that can be in the bottom row?
Answer: (a) 35. (b) 12.

Solution. (a) In any column, there can be no more than one negative number, so the total number of negative numbers is no more than 35. There can be exactly 35 if, for example, the top numbers are $-1, -2, -3, \ldots, -35$, and below them are the numbers $1^{2}, 2^{2}, 3^{2}, \ldots, 35^{2}$ respectively.

(b) We will prove that each number in the bottom row appears no more than 3 times.

- Suppose the number $a0$ appears in the bottom row. If the top number in the column with number $a$ is the square of the bottom number, then the number $a^{2}$ is written above it. If the bottom number is the square of the top number, then above it is written $-\sqrt{a}$ or $\sqrt{a}$. Since all the numbers in the top row are distinct, the number $a$ in the bottom row appears no more than 3 times.

Since each number in the bottom row appears no more than 3 times, and there are 35 numbers in total, there are at least $\frac{35}{3} > 11$, i.e., at least 12 different numbers.

It remains to provide an example of how exactly 12 different numbers can appear in the bottom row. Consider the 12 smallest prime numbers: $2 = p_{1} < p_{2} < \ldots < p_{12}$. Let the number $p_{1}^{2}$ be written in the bottom of the first three columns, and the numbers $-p_{1}, p_{1}$, and $p_{1}^{4}$ be written above them. Similarly, in the next three columns, the number $p_{2}^{2}$ is written in the bottom, and the numbers $-p_{2}, p_{2}$, and $p_{2}^{4}$ are written above them, and so on. In the last two columns, the number $p_{12}^{2}$ is written in the bottom, and the numbers $-p_{12}$ and $p_{12}$ are written above them.