DE-BUNDESWETTBEWERB 2000 Problem 8

Round 2

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if**(a)** $n=2000;$ (b) $n=2001 \ ?$

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Official text: Bundeswettbewerb Mathematik 2000 official paper, page 1.

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