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        riddles >> medium >> solve 4
        
(Message started by: tony123 on Jul 4th, 2007, 6:27am)
Title: solve 4
Post by tony123 on Jul 4th, 2007, 6:27am
http://www.sosmath.com/CBB/latexrender/pictures/a265198aee04e6f1c530d1d393b4985e.gif
Title: Re: solve 4
Post by Sir Col on Jul 4th, 2007, 7:53am
::
[hide]Let S = (61/2-51/2)x + (61/2+51/2)x + (31/2-21/2)x + (31/2+21/2)x

(a-b)-1 = 1/(a-b) = (a+b)/((a-b)(a+b)) = (a+b)/(a2-b2)

If a2 = b2+1, then (a-b)-1 = a+b.

Therefore (61/2-51/2)-x = (61/2+51/2)x and (61/2+51/2)-x = (61/2-51/2)x.

Hence S be the same value for x = -x.

Considering positive x, it is clear that as x decreases S decreases and as x increases S also increases, thus for S = n there is only one solution in positive x.

(u-v)2 + (u+v)2 = 2(u2+v2)

Let a = 61/2, b=51/2, c=31/2, and d=21/2.

So (a-b)2 + (a+b)2 + (c-d)2 + (c+d)2 = 2(6+5) + 2(3+2) = 32.

Hence x = -2,2.[/hide]
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