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Found problems: 1

2011 Gheorghe Vranceanu, 2
Tags: discrete sets , combinatorics
Let $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ be two sequences, the former being a decreasing sequence and the latter being an increasing sequence. All the terms of $ \left( a_i \right)_{1\le i\le n} $ and $ \left( b_i \right)_{1\le i\le n} $ form the set $ \{1,2,3,\ldots ,2n \} . $ Prove that: $$ \left| a_1-b_1 \right| +\left| a_2-b_2 \right| +\cdots +\left| a_n-b_n \right|=n^2 $$