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2018 BAMO 4 a/b+b/c+d/a is integer => abc perfect cube

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August 26, 2019
number theoryalgebraperfect cubeIntegergeometry3D geometry

Problem Statement

(a) Find two quadruples of positive integers (a,b,c,n)(a,b, c,n), each with a different value of nn greater than 33, such that ab+bc+ca=n\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n
(b) Show that if a,b,ca,b, c are nonzero integers such that ab+bc+ca\frac{a}{b} +\frac{b}{c} +\frac{c}{a} is an integer, then abcabc is a perfect cube. (A perfect cube is a number of the form n3n^3, where nn is an integer.)
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