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Found problems: 1

2007 Alexandru Myller, 4
Tags: algebra , induction , finite sequence
Let be a number $ n\ge 2, $ a binary funcion $ b:\mathbb{Z}\rightarrow\mathbb{Z}_2, $ and $ \frac{n^3+5n}{6} $ consecutive integers. Show that among these consecutive integers there are $ n $ of them, namely, $ b_1,b_2,\ldots ,b_n, $ that have the properties: $ \text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right) $ $ \text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1} $