Question 157
Let $q$ be a real number. Margaret wrote 10 pairwise distinct real numbers on a line. Then she adds three lines as follows:
$\triangleright$ on the $2^{\text{nd}}$ line, she writes all numbers of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) reals from the $1^{\text{st}}$ line;
$\triangleright$ on the $3^{\text{rd}}$ line, she writes all numbers of the form $q a b$, where $a$ and $b$ are two (not necessarily distinct) reals 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 (not necessarily distinct) reals from the $2^{\text{nd}}$ line.
Find all reals $q$ such that, whatever the 10 numbers written on the $1^{\text{st}}$ line, every number on the $3^{\text{rd}}$ line is also on the $4^{\text{th}}$ line.
$\triangleright$ on the $2^{\text{nd}}$ line, she writes all numbers of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) reals from the $1^{\text{st}}$ line;
$\triangleright$ on the $3^{\text{rd}}$ line, she writes all numbers of the form $q a b$, where $a$ and $b$ are two (not necessarily distinct) reals 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 (not necessarily distinct) reals from the $2^{\text{nd}}$ line.
Find all reals $q$ such that, whatever the 10 numbers written on the $1^{\text{st}}$ line, every number on the $3^{\text{rd}}$ line is also on the $4^{\text{th}}$ line.