IMO 1984 LL USS64

For a matrix (pij) of the format m imes n with real entries, set

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IMO 1984 LL USS64

Origin: USS

Problem

For a matrix (pij) of the format m \times n with real entries, set ai = n  j=1 pij for i = 1, . . . , m and bj = m  i=1 pij for j = 1, . . . , n. (1) By integering a real number we mean replacing the number with the in- teger closest to it. Prove that integering the numbers ai, bj, pij can be done in such a way that (1) still holds.