On 10/22/07, Fred lunnon <fred.lunnon@gmail.com> wrote:
Find functions f = f(a,b,c), g = g(a,b,c) such that, given any 3 real numbers (a,b,c) /= (0,0,0) with b*b = a*c, it is guaranteed that (f*a + g*b, f*b + g*c) /= (0,0).
I'm not sure I understand exactly what is necessary here, but it sure reminds me of a Putnam problem, A1 from 196... 1962?? not sure exactly the year. The question: Find a real polynomial in two variables whose range is (0, infinity). yeah, that's an OPEN PARENTHESIS there, not [0, infinity). A few blank lines so as not to spoil your fun ... The solution, e.g.: x^2 + (1 - xy)^2 Actually that's not exactly the story: the Putnam proposer thought that no such polynomial existed and the original problem said something like "Does a polynomial exist ..." and only after a student exhibited this polynomial did they realize that the proof that no such polynomial existed. --Joshua Zucker