HMMT 2002 T6

For positive integers $L$ , let $S_{L}=\sum_{n=1}^{L}\lfloor n / 2\rfloor$ . Determine all $L$ for which $S_{L}$ is a square number.

Dataset text NuminaMath olympiads. Quoted with source.

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L=1 \text{ or } L \text{ is even}

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