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        riddles >> putnam exam (pure math) >> Radicals
        
(Message started by: ThudanBlunder on Apr 2nd, 2007, 1:49am)
Title: Radicals
Post by ThudanBlunder on Apr 2nd, 2007, 1:49am
Find all positive numbers a such that 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) is an integer.
Title: Re: Radicals
Post by Barukh on Apr 2nd, 2007, 3:14am
Should a be an integer?
Title: Re: Radicals
Post by Grimbal on Apr 2nd, 2007, 4:59am
That would leave only the numbers 0 to 9 to try.  I would say any number.

"Positive" suggests a can not be complex.  So any positive real number.

There might be extra solutions if the result can be any Gaussian integer.
Title: Re: Radicals
Post by towr on Apr 2nd, 2007, 7:41am
[hide]Isn't the answer all positive numbers? Like last time?

For some integer x, and some y we have [x+ sqrt(y)]3=3+sqrt(a) and [x -  sqrt(y)]3=3-sqrt(a)
x+ sqrt(y) + x -  sqrt(y) = 2 x
[/hide]
Title: Re: Radicals
Post by Grimbal on Apr 2nd, 2007, 10:03am
I don't think you can always find such x and y.

Consider a=0.
3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3+http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3-http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) = 2·3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 and it is not an integer.
Title: Re: Radicals
Post by SMQ on Apr 2nd, 2007, 11:57am
In fact, since a = 0 gives a value of 23http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif3 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/approx.gif 2.88, and the function is monotonic decreasing and always positive, the only integer values it can have are 2 and 1 at the corresponding values of a, right?

--SMQ
Title: Re: Radicals
Post by Barukh on Apr 2nd, 2007, 12:06pm
SMQ, I think you are right.

The problem is to find an expression (I think in radicals) for such a.
Title: Re: Radicals
Post by SMQ on Apr 2nd, 2007, 12:24pm
For x = 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa), I get a = [hide]9 - (x3 - 6)3/(3x)3[/hide], which gives a = [hide]242/27[/hide] at x = 2 and a = [hide]368/27[/hide] at x = 1.

Edit -- derivation:
[hide](1) x = 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)

(2) x3 = 6 + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2)

(3) (x3 - 6)/3 = 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2)

(4) ((x3 - 6)/3)3 = (3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)5(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)4) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)4(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)5) + (3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2

(5) ((x3 - 6)/3)3 = (3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)[6 + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) + 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cdot.gif3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif((3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa)2)]

(6) ((x3 - 6)/3)3 = (9 - a)[x3] (substituting from (2) for the bracketed term)

(7) 9 - a = ((x3 - 6)/(3x))3[/hide]


--SMQ
Title: Re: Radicals
Post by Eigenray on Apr 2nd, 2007, 4:31pm
There's a simpler (or at least easier to read) derivation:

If x = u + v, then
[hide]x3 = u3 + v3 + 3uv x.[/hide]
Substituting then, we have
[hide]x3 = 6 + 3(9-a)1/3 x[/hide],
which is easily solved for a.
Title: Re: Radicals
Post by Barukh on Apr 4th, 2007, 5:52am
:D

How about this variation: 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 + http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) - 3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif(3 - http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifa) is an integer, and a is a rational?
Title: Re: Radicals
Post by SMQ on Apr 4th, 2007, 6:39am
By Eigenray's derivation, x = u - v http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif x3 = [hide]u3 - v3 - 3uvx[/hide] http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif a = [hide]9 - (6 - x3)/(3x)[/sup]3[/sup][/hide] so a is rational (and positive) for [hide]all integer x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ne.gif 0[/hide].

Edit: erm, not quite, as Barukh points out below.  See, this is why I'm a computer programmer and not a mathematician ... other programmers find my off-the-cuff answers convincing. ;)


--SMQ
Title: Re: Radicals
Post by Barukh on Apr 4th, 2007, 10:49am

on 04/04/07 at 06:39:05, SMQ wrote:
x = u - v http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif x3 = [hide]u3 - v3 - 3uvx[/hide][/hide].

What's the value of [hide]u3 - v3[/hide]?


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