### Instruction: For real numbers $a,$ $b,$ and $c,$ the matrix
\[\begin{pmatrix} a & b & c \\ b & c & a \\ c & a & b \end{pmatrix}\]is not invertible.  List all possible values of
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}.\].### Response: To find the possible values of the expression, I need to relate it to the invertibility of the matrix.
One way to do that is to use the fact that a matrix is not invertible if and only if its determinant is zero.
So I will compute the determinant of the matrix and set it equal to zero.
The determinant of a 3 by 3 matrix is given by the formula
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = a(bc - a^2) - b(b^2 - ca) + c(ab - c^2).\]
Expanding and simplifying, I get
\[0 = 3abc - a^3 - b^3 - c^3.\]
This equation looks familiar, it is the same as the identity
\[a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).\]
So I can rewrite the equation as
\[0 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).\]
This means that either $a + b + c = 0$ or $a^2 + b^2 + c^2 - ab - bc - ca = 0.$
I will consider each case separately.
If $a + b + c = 0,$ then I can substitute $a = -b - c$ into the expression and simplify.
I get
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = \frac{-b - c}{b + c} + \frac{b}{-b - c + c} + \frac{c}{-b - c + b}.\]
This simplifies to
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = -1 + \frac{b}{-b} + \frac{c}{-c}.\]
This further simplifies to
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = -1 - 1 - 1 = -3.\]
So one possible value of the expression is -3.
Now I will consider the other case, when $a^2 + b^2 + c^2 - ab - bc - ca = 0.$
This equation can be rewritten as
\[\frac{1}{2}((a - b)^2 + (b - c)^2 + (c - a)^2) = 0.\]
This means that all the squared terms must be zero, so $a = b = c.$
In this case, I can substitute $a = b = c$ into the expression and simplify.
I get
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = \frac{a}{a + a} + \frac{a}{a + a} + \frac{a}{a + a}.\]
This simplifies to
\[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}.\]
So another possible value of the expression is 3/2.
Therefore, the only possible values of the expression are -3 and 3/2.