Determine all possible values of a^2+b^2+c^2
Source: MEMO 2009, problem 2, team competition
October 1, 2009
trigonometryquadraticsalgebra proposedalgebra
Problem Statement
Let , , be real numbers such that for every two of the equations
x^2\plus{}ax\plus{}b\equal{}0, x^2\plus{}bx\plus{}c\equal{}0, x^2\plus{}cx\plus{}a\equal{}0
there is exactly one real number satisfying both of them. Determine all possible values of a^2\plus{}b^2\plus{}c^2.