Problem

Source: Chinese TST 2007 6th quiz P3

Tags: invariant, algebra, polynomial, calculus, function, arithmetic sequence, China



Consider a $ 7\times 7$ numbers table $ a_{ij} = (i^2 + j)(i + j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.