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BCHL85
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#1
But not so easy.
Find x, y, z so that:
4x^2/(1+4x^2) = y
4y^2/(1+4y^2) = z
4z^2/(1+4z^2) = x.
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Gauss
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(Original post by BCHL85)
But not so easy.
Find x, y, z so that:
4x^2/(1+4x^2) = y
4y^2/(1+4y^2) = z
4z^2/(1+4z^2) = x.
x = y = z = 0
x = y = z = 1/2

Galois.
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BCHL85
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(Original post by Galois)
x = y = z = 0
x = y = z = 1/2

Galois.
It's right, but no proving?
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Gauss
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(Original post by BCHL85)
It's right, but no proving?
It's not too difficult. The following method I will show is for x, but as the equations are recursive the method will work for y and z.

Using first two equations, we get

z = 64x^4 / (1 + 8x^2 + 8x^4)

Sub that back into third equation, we get:

x = 16384x^8 / (1 + 16x^2 + 224x^4 + 1280x^6 + 22784x^8)

The highest power of x in this equation is 9, so there are 9 roots.

Solving this equation will give you:

x = 0
x = 1/2 +/- 0i = 1/2
and the other 6 roots are complex numbers (three pairs of complex conjugates).

The following method will work for y and z, so you have

x = y = z = 0
x = y = z = 1/2

I'm prity sure theres another shorter and elegant way, perhaps using matrices, but both methods will get you the same answer.

Galois.
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BCHL85
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(Original post by Galois)
It's not too difficult. The following method I will show is for x, but as the equations are recursive the method will work for y and z.

Using first two equations, we get

z = 64x^4 / (1 + 8x^2 + 8x^4)

Sub that back into third equation, we get:

x = 16384x^8 / (1 + 16x^2 + 224x^4 + 1280x^6 + 22784x^8)

The highest power of x in this equation is 9, so there are 9 roots.

Solving this equation will give you:

x = 0
x = 1/2 +/- 0i = 1/2
and the other 6 roots are complex numbers (three pairs of complex conjugates).

The following method will work for y and z, so you have

x = y = z = 0
x = y = z = 1/2

I'm prity sure theres another shorter and elegant way, perhaps using matrices, but both methods will get you the same answer.

Galois.
I forgot that find x, y, z in R (real solutions)
It's much easier if u show that there's no root if x <> y <> z.
x =4z^2/(1+4z^2) = f(z)
f'(z) = 1 + 8z/(1+4z^2) >0 (cuz z >0)
So if x>y>z -> f(x)>f(y)>f(z) -> y>z>x --> wrong.
Similarly with x>z>y ...
So x = y = z, then just subtitue it to one equation
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