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On the Ratio of Products of Sum of Each Row and Each Column

Source: Problem 4, China National High School Mathematics League 2023

September 10, 2023
algebrainequalities

Problem Statement

Let a=1+10−4a=1+10^{-4}. Consider some 2023×20232023\times 2023 matrix with each entry a real in [1,a][1,a]. Let xix_i be the sum of the elements of the ii-th row and yiy_i be the sum of the elements of the ii-th column for each integer i∈[1,n]i\in [1,n]. Find the maximum possible value of y1y2⋯y2023x1x2⋯x2023\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}} (the answer may be expressed in terms of aa).
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