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Two-pan balance showing the difference

Source: Turkish NMO 1st Round - 2013 - Problem 20

April 19, 2013
combinatorics proposedcombinatorics

Problem Statement

The numbers 1,2,…,20131,2,\dots, 2013 are written on 20132013 stones weighing 1,2,…,20131,2,\dots, 2013 grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in kk weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of kk?
<spanclass=′latex−bold′>(A)</span> 15<spanclass=′latex−bold′>(B)</span> 12<spanclass=′latex−bold′>(C)</span> 10<spanclass=′latex−bold′>(D)</span> 7<spanclass=′latex−bold′>(E)</span> None of above <span class='latex-bold'>(A)</span>\ 15 \qquad<span class='latex-bold'>(B)</span>\ 12 \qquad<span class='latex-bold'>(C)</span>\ 10 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
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