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On the roots of x^3 = x^2 + x + 1 - [Canada MO 1982 - P2]

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September 7, 2011

Problem Statement

If aa, bb and cc are the roots of the equation x3−x2−x−1=0x^3 - x^2 - x - 1 = 0, (i) show that aa, bb and cc are distinct: (ii) show that a1982−b1982a−b+b1982−c1982b−c+c1982−a1982c−a\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a} is an integer.
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