Exercise 6. Let q be a real number. Margaret wrote 10 distinct real numbers in a line. Then she adds three more lines as follows:

$\triangleright$ on the $2^{\text{nd}}$ line, she writes all numbers of the form $a-b$, where $a$ and $b$ are real numbers (not necessarily distinct) from the $1^{\text{st}}$ line;

$\triangleright$ on the $3^{\text{rd}}$ line, she writes all numbers of the form $qab$, where $a$ and $b$ are real numbers (not necessarily distinct) from the $2^{\text{nd}}$ line;

$\triangleright$ on the $4^{\text{th}}$ line, she writes all numbers of the form $a^{2}+b^{2}-c^{2}-d^{2}$, where $a, b, c$ and $d$ are real numbers (not necessarily distinct) from the $2^{\text{nd}}$ line.

Find all real numbers q such that, regardless of the 10 numbers written on the $1^{\text{st}}$ line, each number on the $3^{\text{rd}}$ line is also on the $4^{\text{th}}$ line.
Solution to Exercise 6: We will say that a real number \( q \) is good if it has the property required in the statement. First, it is clear that if \( q \) is good, then \( -q \) is also good, and vice versa. Moreover, \( q = 0 \) is obviously good. We are therefore looking for good real numbers \( q > 0 \), if there are any.

Let \( \lambda \) be a large real number and let \( \varepsilon = 1 / \lambda \). Suppose Margaret has written ten numbers \( x_{1}, \ldots, x_{10} \) such that \( 0 \leqslant x_{1} < \ldots < x_{8} \leqslant \varepsilon \), \( x_{9} = 1 \), and \( x_{10} = \lambda \). Then the second line contains the numbers \( \pm \lambda, \pm (\lambda - 1) \), and \( \pm 1 \), as well as elements of the set \( \Omega_{2} = [0, 1] \cup [\lambda - \varepsilon, \lambda] \) and their opposites. In particular, Margaret writes the real numbers \( q \lambda^{2} \) and \( q \lambda \) on the third line.

We then consider a number from the fourth line, of the form \( z = a^{2} + b^{2} - c^{2} - d^{2} \), and such that \( z \geqslant 0 \).

![](https://cdn.mathpix.com/cropped/2024_06_04_e6ed491384f68a8fcb15g-4.jpg?height=62&width=1743&top_left_y=197&top_left_x=191)
to the set \( \Omega_{4} = \{0, \lambda^{2}, 2 \lambda^{2}\} + \{-4 \lambda, -2 \lambda, 0, 2 \lambda, 4 \lambda\} + [-4, 4] \).

If \( q \) is good, then \( q \lambda^{2} \) and \( q \lambda \) belong to \( \Omega_{4} \), so by letting \( \lambda \) tend to \( +\infty \), we deduce respectively that \( q \in \{1, 2\} \) and \( q \in \{2, 4\} \). This shows that only \( q = 2 \) is potentially good.

Conversely, if \( q = 2 \), every number on the third line is of the form \( 2(a - b)(c - d) \) with \( a, b, c, d \) being real numbers from the first line. Since \( 2(a - b)(c - d) = (a - d)^{2} + (b - c)^{2} - (a - c)^{2} - (b - d)^{2} \), this number also appears on the fourth line. Therefore, the good real numbers are \( -2, 0 \), and \( 2 \).