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Part of 2009 Middle European Mathematical Olympiad

Problems(1)

Determine all possible values of a^2+b^2+c^2

Source: MEMO 2009, problem 2, team competition

a

b

c

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.

trigonometryquadraticsalgebra proposedalgebra