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Turkish NMO First Round - 2000 P-35 (Combinatorics)

Source:

July 8, 2012

Problem Statement

If every kk-element subset of S={1,2,,32}S=\{1,2,\dots , 32\} contains three different elements a,b,ca,b,c such that aa divides bb, and bb divides cc, kk must be at least ?
<spanclass=latexbold>(A)</span> 17<spanclass=latexbold>(B)</span> 24<spanclass=latexbold>(C)</span> 25<spanclass=latexbold>(D)</span> 29<spanclass=latexbold>(E)</span> None <span class='latex-bold'>(A)</span>\ 17 \qquad<span class='latex-bold'>(B)</span>\ 24 \qquad<span class='latex-bold'>(C)</span>\ 25 \qquad<span class='latex-bold'>(D)</span>\ 29 \qquad<span class='latex-bold'>(E)</span>\ \text{None}