I'm having real trouble answering a question posted online as appearing in a 1971 A level exam (as part of a discussion of whether exams have got easier over the years - which this example would seem to confirm!). It can be found at bit.ly/mZoyP8, and goes like this:
"If a + b + c=a2+b2+c2=a3+b3+c3=2, find by considering values of (a+b+c)2 and (a+b+c)3, or otherwise, the values of (i) ab+bc+ca, (ii) abc.
Hence find the equation whose roots are a, b and c."
According to my calculations, ab+bc+ca = 1 & abc = -2/3. But I've been struggling for hours to complete the final step. You end up with a set of three simultaneous (?) equations (the two implied by these answers, plus the original "a+b+c = 2") from which it seems impossible to derive a (presumably?) cubic equation that will give have the three roots.
Anyone got any ideas?