Question 142
Let $q \in \mathbb{R}$. There are 10 distinct real numbers on the blackboard. Alex writes the following three lines of numbers:
1. In the first line Alex writes down every number of the form $a-b$, where $a, b$ are two (not necessarily distinct) numbers from the board;
2. In the second line Alex writes down every number of the form $qab$, where $a, b$ are two (not necessarily distinct) numbers from the first line;
3. In the third line Alex writes down every number of the form $a^2 + b^2 - c^2 - d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
Determine all values of $q$ such that, regardless of the numbers on the board, every number in the second line is also a number in the third line.
(IMO-2017 Shortlist, Problem A2)
1. In the first line Alex writes down every number of the form $a-b$, where $a, b$ are two (not necessarily distinct) numbers from the board;
2. In the second line Alex writes down every number of the form $qab$, where $a, b$ are two (not necessarily distinct) numbers from the first line;
3. In the third line Alex writes down every number of the form $a^2 + b^2 - c^2 - d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
Determine all values of $q$ such that, regardless of the numbers on the board, every number in the second line is also a number in the third line.
(IMO-2017 Shortlist, Problem A2)