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Squares in a board - Greatest possible sums of the numbers

Source: IMO LongList 1970 - P57

May 24, 2011
combinatorics proposedcombinatorics

Problem Statement

Let the numbers 1,2,…,n21, 2, \ldots , n^2 be written in the cells of an n×nn \times n square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the kthk^{th} row? (kk a positive integer, 1≤k≤n1 \leq k \leq n.)
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