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IMO ShortList 2002, algebra problem 5

Source: IMO ShortList 2002, algebra problem 5

September 28, 2004
linear algebraalgebrasystem of equationsIMO Shortlist

Problem Statement

Let nn be a positive integer that is not a perfect cube. Define real numbers a,b,ca,b,c by a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-}\kern1.5pt, where [x][x] denotes the integer part of xx. Prove that there are infinitely many such integers nn with the property that there exist integers r,s,tr,s,t, not all zero, such that ra+sb+tc=0ra+sb+tc=0.
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