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Let n be a positive integer that is not a perfect cube. Define real numbers a,b,c by

a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-[b]}\kern1.5pt,

where [x] denotes the integer part of x. Prove that there are infinitely many such integers n with the property that there exist integers r,s,t, not all zero, such that ra+sb+tc=0.

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