## Description

In a rectangular array of nonnegative real numbers with r rows and c columns, and with r < c, each column contains at least one positive element.

Prove: There exists a positive element for which the sum of the elements of the intersecting row (its row sum) is larger than the sum of the elements of the intersecting column (its column sum).

Note: This is a task for advanced high school math students. **The solution should be worked out and explained in detail, preferably using high school math. **

## Explanation & Answer

Attached.

1

Running Head: ALGEBRA

Algebra

First Middle Last

Name of Institution

2

ALGEBRA

Solution

Suppose that 𝑟 does not equal 𝑐 and 𝑟 < 𝑐. We move to the next step by applying strong

induction. If 𝑎 < 𝑟 and 𝑏 < 𝑐, we will assume that the statement is valid. The base case is

obviously valid, where either 𝑟 = 1 or 𝑐 = 1. Let’s now contemplate a bipartite graph between

the set of rows and the set of columns. The bipartite graph intersects in a positive element when

there ...

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