Wednesday, 2 October 2013

Pairwise Different

101 distinct numbers are chosen among the integers between 0 and 1000. Prove that, among the absolute values of their pairwise di fferences, there are ten di fferent numbers not exceeding 100.
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Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, ...