IMO Shortlist 2002 problem N1

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with

$x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?$

Izvor: Međunarodna matematička olimpijada, shortlist 2002

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