IMO 2017 Shortlist A2
Let q be a real number. Gugu has a napkin with ten distin t real numbers written on it, and he writes the following thre...
Category: Algebra
Problem
Let q be a real number. Gugu has a napkin with ten distin t real numbers written on it, and he writes the following three lines of real numbers on the bla kboard: • In the rst line, Gugu writes down every number of the form a´b, where a and b are two (not ne essarily distin t) numbers on his napkin. • In the se ond line, Gugu writes down every number of the form qab, where a and b are two (not ne essarily distin t) numbers from the rst line. • In the third line, Gugu writes down every number of the form a2 ` b2 ´ c2 ´ d2 , where a,b,c,d are four (not ne essarily distin t) numbers from the rst line. Determine all values of q su h that, regardless of the numbers on Gugu's napkin, every number in the se ond line is also a number in the third line. (Austria)