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Coining a Grid

Source: Canada Repechage 2022/8 CMOQR

March 19, 2022
repechageinequalities

Problem Statement

Let {m,n,k}\{m, n, k\} be positive integers. {k}\{k\} coins are placed in the squares of an m×nm \times n grid. A square may contain any number of coins, including zero. Label the {k}\{k\} coins C1,C2,⋅⋅⋅CkC_1, C_2, · · · C_k. Let rir_i be the number of coins in the same row as CiC_i, including CiC_i itself. Let sis_i be the number of coins in the same column as CiC_i, including CiC_i itself. Prove that ∑i=1k1ri+si≤m+n4\sum_{i=1}^k \frac{1}{r_i+s_i} \leq \frac{m+n}{4}
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