Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$.

D. Bazylev
1. **Check for \( n = 1 \):**
   \[
   x^1 + y^1 + z^1 = x + y + z
   \]
   Given \( x + y + z = 0 \), it follows that:
   \[
   x + y + z = 0
   \]
   Hence, \( x^1 + y^1 + z^1 \) is constant (specifically, it is always 0).

2. **Check for \( n = 3 \):**
   Using the identity:
   \[
   x^3 + y^3 + z^3 = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) + 3xyz
   \]
   Given \( x + y + z = 0 \) and \( xyz = 1 \), we have:
   \[
   x^3 + y^3 + z^3 = 0 \cdot (x^2 + y^2 + z^2 - xy - yz - zx) + 3xyz = 3 \cdot 1 = 3
   \]
   Hence, \( x^3 + y^3 + z^3 \) is constant (specifically, it is always 3).

3. **Check for \( n = 2 \):**
   We need to determine if \( x^2 + y^2 + z^2 \) is constant. Using the identity:
   \[
   (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
   \]
   Given \( x + y + z = 0 \), we have:
   \[
   0 = x^2 + y^2 + z^2 + 2(xy + yz + zx)
   \]
   Thus:
   \[
   x^2 + y^2 + z^2 = -2(xy + yz + zx)
   \]
   Since \( xy + yz + zx \) is not constant for all \( x, y, z \) satisfying \( x + y + z = 0 \) and \( xyz = 1 \), \( x^2 + y^2 + z^2 \) is not constant.

4. **Check for \( n > 3 \):**
   Assume \( x \) is fixed and solve for \( y \) and \( z \):
   \[
   y, z = \frac{-x^{3/2} \pm \sqrt{x^3 - 4}}{2\sqrt{x}}
   \]
   Consider the function:
   \[
   f(x) = x^n + \left( \frac{-x^{3/2} + \sqrt{x^3 - 4}}{2\sqrt{x}} \right)^n + \left( \frac{-x^{3/2} - \sqrt{x^3 - 4}}{2\sqrt{x}} \right)^n
   \]
   For \( n > 3 \), as \( x \to \infty \):
   \[
   \left( \frac{-x^{3/2} \pm \sqrt{x^3 - 4}}{2\sqrt{x}} \right)^n \approx \left( \frac{-x^{3/2}}{2\sqrt{x}} \right)^n = \left( -\frac{x}{2} \right)^n
   \]
   Thus:
   \[
   f(x) \approx x^n + 2 \left( -\frac{x}{2} \right)^n = x^n + 2 \left( -\frac{x}{2} \right)^n
   \]
   Since \( \left( -\frac{x}{2} \right)^n \) grows without bound as \( x \to \infty \), \( f(x) \) cannot be constant.

Conclusion:
The only values of \( n \) for which \( x^n + y^n + z^n \) is constant for all \( x, y, z \in \mathbb{R} \) such that \( x + y + z = 0 \) and \( xyz = 1 \) are \( n = 1 \) and \( n = 3 \).

The final answer is \( \boxed{ n = 1 } \) and \( n = 3 \).