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(1/a+1/b+1/c)(a+b+c-2) if a+b+c=a^2+b^2+c^2=a^3+b^3+c^3

Source: 2011 Belarus TST 2.4

November 8, 2020
algebrasystem of equationsBelarus

Problem Statement

Given nonzero real numbers a,b,c with a+b+c=a2+b2+c2=a3+b3+c3a+b+c=a^2+b^2+c^2=a^3+b^3+c^3. (∗*) a) Find (1a+1b+1b)(a+b+c−2)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2) b) Do there exist pairwise different nonzero a,b,ca,b,c satisfying (∗*)?
D. Bazylev
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