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.
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Running Head: ALGEBRA
Algebra
First Middle Last
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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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