## March 2022

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In the Math Corner of my March Diary I offered this:

Letaandbbe positive integers witha≥b. Prove that just one of the following things must be true.

Eithera/(a+b) + (a+b)/a> √5

ora/b+b/a> √5.

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**• Solution**

If you stare hard there you will see that we are dealing with the function

f(x) =x+ 1/x.

Let's writeuforb/a. Then the facts thataandbare positive integers anda≥btell us thatuis a positive rational number less than or equal to 1.

With this notation in our hands we can rewrite the thing we're required to prove as:

For a positive rational numberuless than or equal to 1, eitherf(u) > √5 orf(1 +u) > √5.

(Just pause to convince yourself this is right …)

First let's ask: When isf(x) = √5? That's easy; a simple quadratic equation whose roots turn out to be (√5 ± 1)/2.

Now (√5 + 1)/2 is of course the Golden Ratioφ, equal to 1.618033988749894848204586834365638117720 … The other root, (√5 − 1)/2, isφ− 1; and it is also 1/φ(that's what makes the Golden Ratio golden).

At this point it's helpful to have a graph off(x), so I've drawn one. The red curve is the graph off(x); the blue horizontal line isy= √5; the purple vertical line isx=φ; the green vertical line isx=φ− 1.

Since we know thatuis positive and less than or equal to 1, the only part of the graph that can represent values ofuthat interest us is the vertical strip fromx= 0 (exclusive) tox= 1 (inclusive); and since we know thatuis a rational number whileφis irrational, we also know that the green vertical line cannot representu.

In that strip,f(u) > √5 whenuis between zero (exclusive) and the green line atφ− 1 (exclusive).

But what ifuis between the green line atφ− 1 (exclusive) andx= 1 (inclusive)? Well, then 1 +uis between the purple line atφ(exclusive) andx= 2 (inclusive). For those values, indeedf(1 +u) > √5.

So for all the valuesucan take under the conditions given onaandb, eitherf(u) > √5 orf(1 +u) > √5.

And sinceucannot simultaneously be to both the leftandthe right of the vertical green line, andf(x) is a single-valued function, that "or" is exclusive: just one of the two things is true. QED.