Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

Find four numbers

Find four numbers A, B, C, D

so that 0 < A< B < C < D and

.

Last edited by tony123 (2008-08-04 13:27:00)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Find four numbers

Edit: And after investigating a bit, it looks like working sets can be generated.

Edit2: And I've just convinced myself that that always works. The proof's not that hard, but it is quite messy so I won't put it here.

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Why did the vector cross the road?
It wanted to be normal.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Find four numbers

Jane had a post here earlier saying something like:

It got deleted for some reason but I have no idea why. The answer is brilliant and flawless, as far as I can see, and gives an extra degree of freedom to my one.

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Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Find four numbers

The answer is brilliant and flawless, as far as I can see, and gives an extra degree of freedom to my one.

Was it, really?

I thought I had posted too early because I had forgotten that you need C > B as well. Anyway, I’m now satisfied that this can be done if you choose A such that

.

If

then you have

and if you take

you would also have

.

And

.

So, yes. You can always choose A, B and C such that

.

And the last thing is to make sure is, of course, to choose A and B such that

.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Find four numbers

Oh right, you were working with the reals.
I was thinking with naturals, where neither of those concerns affect your solution (B < 2AB and B < B² - A² are both always true when A<B, and 2AB is never equal to B² - A²).

Unless I've made another embarrassing mistake, anyway.

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Why did the vector cross the road?
It wanted to be normal.