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Show that $ b_1 + b_2 + ... + b_n$ =

Source: India National Olympiad 2002, Problem 6

October 10, 2005
combinatorics unsolvedcombinatorics

Problem Statement

The numbers 1,2,31, 2, 3, …\ldots, n2n^2 are arranged in an n×nn\times n array, so that the numbers in each row increase from left to right, and the numbers in each column increase from top to bottom. Let aija_{ij} be the number in position i,ji, j. Let bjb_j be the number of possible values for ajja_{jj}. Show that b1+b2+⋯+bn=n(n2−3n+5)3. b_1 + b_2 + \cdots + b_n = \frac{ n(n^2-3n+5) }{3} .
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