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100 students study in groups of two

Source: Turkish NMO 1st Round - 2013 - Problem 12

April 20, 2013
combinatorics proposedcombinatorics

Problem Statement

In the morning, 100100 students study as 5050 groups with two students in each group. In the afternoon, they study again as 5050 groups with two students in each group. No matter how the groups in the morning or groups in the afternoon are established, if it is possible to find nn students such that no two of them study together, what is the largest value of nn?
<spanclass=latexbold>(A)</span> 42<spanclass=latexbold>(B)</span> 38<spanclass=latexbold>(C)</span> 34<spanclass=latexbold>(D)</span> 25<spanclass=latexbold>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 42 \qquad<span class='latex-bold'>(B)</span>\ 38 \qquad<span class='latex-bold'>(C)</span>\ 34 \qquad<span class='latex-bold'>(D)</span>\ 25 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}