ISL 2002 N1

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with $x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?$

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Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.