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Chinese TST 2008 P5

Source:

April 4, 2008
functioninequalitiesCauchy Inequalityalgebra proposedalgebra

Problem Statement

For two given positive integers m,n>1 m,n > 1, let aij(i=1,2,⋯ ,n,  j=1,2,⋯ ,m) a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m) be nonnegative real numbers, not all zero, find the maximum and the minimum values of f f, where f=n∑i=1n(∑j=1maij)2+m∑j=1m(∑i=1naij)2(∑i=1n∑j=1maij)2+mn∑i=1n∑j=1maij2. f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}.
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