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S has at most six elements

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August 30, 2010

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Problem Statement

Let

a, b, c

be positive real numbers and

p, q, r

complex numbers. Let

S

be the set of all solutions

(x, y, z)

in

\mathbb C

of the system of simultaneous equations

ax + by + cz = p,

ax2 + by2 + cz2 = q,

ax3 + bx3 + cx3 = r.

Prove that

S

has at most six elements.

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