Find all sequences satisfying two conditions
Source: IMO Shortlist 2007, C1, AIMO 2008, TST 1, P1
July 13, 2008
combinatoricsSequenceIMO Shortlistinequality systeminduction
Problem Statement
Let be an integer. Find all sequences a_1, a_2, \ldots a_{n^2 \plus{} n} satisfying the following conditions:
\text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n;
\text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n.
Author: Dusan Dukic, Serbia