Problem

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Tags: combinatorics



Alberto chooses 2022 integers a1,a2,,a2022 (not necessarily positive and not necessarily distinct) and places them on a 2022×2022 table such that in the (i,j) cell is the number ak, with k=max, as shown in figure (in which, for a better readability, we have denoted a_{2022} with a_n). Barbara does not know the numbers Alberto has chosen, but knows how they are displaced in the table. Given a positive integer k, with 1\leq k\leq 2022, Barbara wants to determine the value of a_k (and she is not interested in determining the values of the other a_i's with i\neq k). To do so, Barbara is allowed to ask Alberto one or more questions, in each of which she demands the value of the sum of the numbers contained in the cells of a "path", where with the term "path" we indicate a sorted list of cells with the following characteristics: • the path starts from the top left cell and finishes with the bottom right cell, • the cells of the path are all distinct, • two consecutive cells of the path share a common side. Determine, as k varies, the minimum number of questions Barbara needs to find a_k.


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