1. ## Find x1,x2,x3,x4,x5

Find the real numbers $x_1 ,x_2 ,x_3 ,x_4 ,x_5$ $
$
solutions this system, when m is the real parametr :

$\left\{ \begin{array}{l}
x_5 + x_2 = mx_1 \\
x_1 + x_3 = mx_2 \\
x_2 + x_4 = mx_3 \\
x_3 + x_5 = mx_4 \\
x_4 + x_1 = mx_5 \\
\end{array} \right.$

2. Originally Posted by dhiab
Find the real numbers $x_1 ,x_2 ,x_3 ,x_4 ,x_5$ $
$
solutions this system, when m is the real parametr :

$\left\{ \begin{array}{l}
x_5 + x_2 = mx_1 \\
x_1 + x_3 = mx_2 \\
x_2 + x_4 = mx_3 \\
x_3 + x_5 = mx_4 \\
x_4 + x_1 = mx_5 \\
\end{array} \right.$
There is an obvious solution $(x_1,x_2,x_3,x_4,x_5) = (0,0,0,0,0)$. For all but three values of m, that will be the only solution. The exceptional values of m are $m=2$ and $m= \tfrac12(-1\pm\sqrt5)$.

If $m=2$, the solution is $(x_1,x_2,x_3,x_4,x_5) = s(1,1,1,1,1)$ (for any real number s).

If $m= \tfrac12(-1\pm\sqrt5)$ then m satisfies the equation $m^2+m-1=0$ and the solution is $(x_1,x_2,x_3,x_4,x_5) = s(-1,-m,m,1,0) + t(m,-m,-1,0,1)$ (for any real numbers s and t).

3. dhiab, your problem post is getting harder . . . .

4. Originally Posted by pacman
dhiab, your problem post is getting harder . . . .