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Turkey NMO 2006 1st Round - P36 (Combinatorics)

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February 3, 2013

Problem Statement

In an exam with nn problems where nn is a positive integer, each problem was answered by at least one student. Each student answered an even number of problems. Any two students answered an even number of problems in common. What is the number of values that nn cannot take?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 4<spanclass=′latex−bold′>(C)</span> 5<spanclass=′latex−bold′>(D)</span> Infinitely many<spanclass=′latex−bold′>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ 4 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ \text{Infinitely many} \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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