MathDB

Inequality with ordering

Source: 2021 China TST, Test 1, Day 1 P1

March 17, 2021

inequalities

Problem Statement

Given positive integers

m

and

n

. Let

a_{i,j} ( 1 \le i \le m, 1 \le j \le n)

be non-negative real numbers, such that

a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j}

holds for all

1 \le i \le m

and

1 \le j \le n

. Denote

X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},

Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.

Prove that

\prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.

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