Given six real numbers $a$, $b$, $c$, $x$, $y$, $z$ such that $0 < b-c < a < b+c$ and $ax + by + cz = 0$. 

What is the sign of the sum $ayz + bzx + cxy$ ?
Given the conditions \(0 < b - c < a < b + c\) and \(ax + by + cz = 0\), we need to determine the sign of the expression \(ayz + bzx + cxy\).

1. **Interpret the given conditions**:
   - The inequality \(0 < b - c < a < b + c\) implies that \(a\), \(b\), and \(c\) can be the sides of a triangle. This is because the triangle inequality is satisfied.
   - The equation \(ax + by + cz = 0\) suggests that the point \(P(x, y, z)\) lies on the line at infinity in the context of trilinear coordinates.

2. **Normalize the expression**:
   - Consider the expression \(\frac{ayz + bzx + cxy}{x^2 + y^2 + z^2}\). We need to show that this expression is negative. Since \(x^2 + y^2 + z^2 > 0\) for any non-zero \(x, y, z\), the sign of \(\frac{ayz + bzx + cxy}{x^2 + y^2 + z^2}\) is the same as the sign of \(ayz + bzx + cxy\).

3. **Analyze the expression**:
   - We need to show that \(ayz + bzx + cxy < 0\) for all \(x, y, z\) satisfying \(ax + by + cz = 0\).

4. **Consider specific values**:
   - Let's consider specific values for \(x, y, z\) that satisfy \(ax + by + cz = 0\). For instance, let \(x = 0\), \(y = -c\), and \(z = b\). Substituting these into the equation \(ax + by + cz = 0\), we get:
     \[
     a \cdot 0 + b \cdot (-c) + c \cdot b = 0 \implies -bc + bc = 0
     \]
     This satisfies the equation.

5. **Evaluate the expression**:
   - Substitute \(x = 0\), \(y = -c\), and \(z = b\) into \(ayz + bzx + cxy\):
     \[
     ayz + bzx + cxy = a \cdot (-c) \cdot b + b \cdot 0 \cdot b + c \cdot 0 \cdot (-c) = -abc
     \]
     Since \(a, b, c > 0\), we have \(-abc < 0\).

6. **Generalize the result**:
   - The expression \(\frac{ayz + bzx + cxy}{x^2 + y^2 + z^2}\) does not change sign as \(P(x, y, z)\) moves along the line at infinity. Since we have shown that it is negative for one specific set of values, it must be negative for all values satisfying \(ax + by + cz = 0\).

Conclusion:
\[
\boxed{\text{The expression } ayz + bzx + cxy \text{ is negative.}}
\]