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Romania counting and probability
Problem
The cells of a matrix are coloured using colours. A colour is called dominant on a row (or a column) if there are at least cells of this colour on that row (or column). A cell is called extremal if its colour is dominant both on its row, and its column. Find all for which there is a colouring with no extremal cells.
Solution
The answer is that there exist such colourings for (almost trivial), and (following a detailed and tedious case analysis). For one must exhibit a colouring with no extremal cells (again a very tedious labor). The colourings with no extremal cells for are built inductively, using a beautifully symmetric pattern based on the model for (this being the elegant combinatorial part of the problem).
A detailed proof for this problem can be found in no. 4/2006, pages 22-25, of the on-line magazine Mathematical Reflections http://reflections.awesomemath.org/2006_4/2006_4_solutions.pdf.
A detailed proof for this problem can be found in no. 4/2006, pages 22-25, of the on-line magazine Mathematical Reflections http://reflections.awesomemath.org/2006_4/2006_4_solutions.pdf.
Final answer
All integers n ≥ 2
Techniques
Coloring schemes, extremal argumentsInduction / smoothing