How to solve this system of nonlinear equations? How to solve these equations for $a$, $b$, $c$ and $x$?
I have the following:
\begin{align}
1 &=  2a+b+c\\
a &= (a+b)x + 0.25(a+c)\\
a&=(a+c)(1-x)\\
b&=a(1-x)+c(x-0.25)\\
c&=b(1-x)+a(x-0.25)
\end{align}
I tried, but ended circular at the point I started. Can someone help me?

Using Grobner basis in maxima gives:
grobner_basis([2*a+b+c-1,(a+b)x+(a+c)/4-a,(a+c)(1-x)-a,a*(1-x)+c*(x-1/4)-b,b*(1-x)+a*(x-1/4)-c]);
(- 44) x + (- 144) a^2  + 213 a + (- 36), 
(- 11) c + (- 48) a^2  + 60 a + (- 12),
(- 11) b + 48 a^2  + (- 82) a + 23, 
(- 144) a^3  + 381 a^2  + (- 312) a + 64
The last equation is a cubic in "a" alone.
The second last solves for b given a.
The third last solves for c in a.
The top equation solves for x in a.
Grobner Basis and Buchberger's algorithm explain how this works.
regards arthur

lower case x
grobner_basis([2*A+B+C-1,(A+B)x+(A+C)/4-A,(A+C)(1-x)-A,A*(1-x)+C*(x-1/4)-B,B*(1-x)+A*(x-1/4)-C]);
(- 3) C + 4 x^2 ,
(- 3) B + 4 x^2  + (- 8) x + 3, 
(- 3) A + (- 4) x^2 + 4 x,
(- 8) x^3  + 13 x^2  + (- 12) x + 3
