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Page by Pat Lachapelle. Last updated: 03 Dec 2019

1985 Putnam Question A-2: Solution Part 2

Overview

In the previous installment of this series, we discussed the problem and established there must be a maximum value of the expression:

`\frac{A(R)+A(S)}{A(T)}`

Generalization is a tool sometimes used by Mathematicians to solve complicated problems. In this section we will discuss and explain generalization in depth. Then we will utilize it in taking the first substantial step in solving the problem.

Generalization

Generalization is a very useful concept in Mathematics that can help solve problems of almost any sort. The primary concept is rather simple and straightforward: if something is true for all possible cases, it must be true for any specific case.

It is useful because certain things that may be difficult or impossible to solve for specific cases may be easier to solve for more general scenarios.

For example, evaluate:

`10xx1`

The answer is obviously `10`

Now consider: `10xx2`

Just as easy, the answer is `20`

A clear pattern arises here. To generalize we can say that to multiply any number by `10`, just add a zero to that number. Pretty easy right?

So what is

`5,821,693xx10`?

Solving this by hand might take some time, but we can use our generalization to find the answer to be `58,216,930.`

This was a very simple generalization, one that you're most likely well aware of, but the concept applies to far more complicated examples.

Generalizing our Problem

Recall that we are attempting to generalize our problem, which involves precisely two rectangles inscribed within a single, acute triangle (which may, or may not be, the same height).

So how can we generalize this problem?

We must keep the level to which we generalize somewhat reasonable. For instance, it is probably unrealistic that we can generalize the scenario to fit any shape, or even any polygon for that matter. Triangles have many unique properties (some of which we will discuss in the next section), that will prove to be essential to our solving of this problem.

So, if we require large polygon to be a triangle, specifically an acute triangle, what can we generalize?

Well we can still generalize for the number of rectangles inscribed within the triangle. For instance, what if there were three rectangles within the triangle?

T
R1
R2
R3

How about four?

T
R1
R2
R3
R4

What if we generalized the problem so that there are n rectangles inscribed within the acute triangle. In this scenario, we can let the rectangles be called `R_1, R_2, R_3`, and so forth, all the way up to `R_n.`

T
R1
R2
R3
R4
R5
Rn

This generalizes the problem in a way that will make it easier for us to solve the problem in the next section.

Summary

What we did in this section may not seem all that important. But upon finishing the series, you will probably look back at this section as the one that laid the foundation for the path that will take us to the final answer. Understanding this concept of generalization can be quite difficult, making it this far is an accomplishment in and of itself. Thanks for reading, I hope you've enjoyed this series thus far.

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