IMO 1970 LL USS57

Let the numbers 1, 2, . . . , n2 be written in the cells of an n imes n

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IMO 1970 LL USS57

Origin: USS

Problem

Let the numbers 1, 2, . . . , n2 be written in the cells of an n \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 kth row? (k a positive integer, 1 \leqk \leqn.)