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ASU 026 All Russian MO 1962 10.5 table

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June 17, 2019
combinatoricstable

Problem Statement

Given positive numbers a1,a2,...,am,b1,b2,...,bna_1, a_2, ..., a_m, b_1, b_2, ..., b_n. Is known that a1+a2+...+am=b1+b2+...+bn.a_1+a_2+...+a_m=b_1+b_2+...+b_n. Prove that you can fill an empty table with mm rows and nn columns with no more than (m+n−1)(m+n-1) positive number in such a way, that for all i,ji,j the sum of the numbers in the ii-th row will equal to aia_i, and the sum of the numbers in the jj-th column -- to bjb_j.
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