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Dividing 77 stones into k groups

Source: Turkish NMO 1st Round - 2013 - Problem 24

April 20, 2013
ceiling functioninequalitiescombinatorics proposedcombinatorics

Problem Statement

7777 stones weighing 1,2,,771,2,\dots, 77 grams are divided into kk groups such that total weights of each group are different from each other and each group contains less stones than groups with smaller total weights. For how many k{9,10,11,12}k\in \{9,10,11,12\}, is such a division possible?
<spanclass=latexbold>(A)</span> 4<spanclass=latexbold>(B)</span> 3<spanclass=latexbold>(C)</span> 2<spanclass=latexbold>(D)</span> 1<spanclass=latexbold>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 4 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 1 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}