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April 21, 2009
pigeonhole principle

Problem Statement

Points on a square with side length c c are either painted blue or red. Find the smallest possible value of c c such that how the points are painted, there exist two points with same color having a distance not less than 5 \sqrt {5}.
<spanclass=′latex−bold′>(A)</span> 102<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> 22<spanclass=′latex−bold′>(E)</span> None<span class='latex-bold'>(A)</span>\ \frac {\sqrt {10} }{2} \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ \sqrt {5} \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt {2} \qquad<span class='latex-bold'>(E)</span>\ \text{None}
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