IMO Shortlist 2017 problem A2


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3. listopada 2019.
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Let q be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: In the first line, Gugu writes down every number of the form a-b, where a and b are two (not necessarily distinct) numbers on his napkin. In the second line, Gugu writes down every number of the form qab, where a and b are two (not necessarily distinct) numbers from the first line. In the third line, Gugu 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 Gugu's napkin, every number in the second line is also a number in the third line.

Izvor: https://www.imo-official.org/problems/IMO2017SL.pdf