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Which integers are (a/b+b/c+c/a) and (b/a+a/c+c/b)?

Source: Pan African MO 2006 Q2

May 1, 2013

VietaalgebrapolynomialRational Root Theoremnumber theory unsolvednumber theory

Problem Statement

Let

a, b, c

be three non-zero integers. It is known that the sums

\frac{a}{b}+\frac{b}{c}+\frac{c}{a}

and

\frac{b}{a}+\frac{c}{b}+\frac{a}{c}

are integers. Find these sums.