Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$
$f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0$
$f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0$
$...$
$f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0$
1. **Insert \( x = y \) in the first equation:**
   \[
   f_1(x) - f_2(x)f_2(x) + f_1(x) = 0 \implies 2f_1(x) = f_2(x)^2
   \]
   This implies:
   \[
   f_2(x)^2 = 2f_1(x)
   \]

2. **Consider the expression \((f_1(x) + f_1(y))^2\):**
   \[
   (f_1(x) + f_1(y))^2 = (f_2(x)f_2(y))^2 = 4f_1(x)f_1(y)
   \]
   This implies:
   \[
   (f_1(x) - f_1(y))^2 = 0 \implies f_1(x) = f_1(y)
   \]
   Therefore, \( f_1(x) \) is a constant function. Let \( f_1(x) = C_1 \).

3. **Substitute \( f_1(x) = C_1 \) into the first equation:**
   \[
   2C_1 = f_2(x)^2
   \]
   Since \( C_1 \) is constant, \( f_2(x) \) must also be constant. Let \( f_2(x) = C_2 \).

4. **Repeat the process for the second equation:**
   \[
   f_2(x^2) - f_3(x)f_3(y) + f_2(y^2) = 0
   \]
   Substituting \( f_2(x) = C_2 \):
   \[
   C_2 - f_3(x)f_3(y) + C_2 = 0 \implies 2C_2 = f_3(x)f_3(y)
   \]
   Since \( C_2 \) is constant, \( f_3(x) \) must also be constant. Let \( f_3(x) = C_3 \).

5. **Continue this process for all \( f_i \):**
   \[
   f_i(x^i) - f_{i+1}(x)f_{i+1}(y) + f_i(y^i) = 0
   \]
   Substituting \( f_i(x) = C_i \):
   \[
   2C_i = f_{i+1}(x)f_{i+1}(y)
   \]
   Since \( C_i \) is constant, \( f_{i+1}(x) \) must also be constant. Let \( f_{i+1}(x) = C_{i+1} \).

6. **Determine the constants \( C_i \):**
   \[
   2C_i = C_{i+1}^2
   \]
   For \( i = 1 \):
   \[
   2C_1 = C_2^2
   \]
   For \( i = 2 \):
   \[
   2C_2 = C_3^2
   \]
   Continue this process until \( i = n \):
   \[
   2C_{n-1} = C_n^2
   \]
   And for the last equation:
   \[
   2C_n = C_1^2
   \]

7. **Solve the system of equations:**
   \[
   C_1^2 = 2C_n
   \]
   \[
   2^{2^{i-1}-1} C_1 = C_i^{2^{i-1}}
   \]
   Substituting \( n \) for \( i \):
   \[
   C_1 = 2 \quad \text{or} \quad C_1 = 0
   \]

8. **Check the solutions:**
   - If \( C_1 = 0 \):
     \[
     0 = C_2 = C_3 = \dots = C_n
     \]
   - If \( C_1 = 2 \):
     \[
     2 = C_2 = C_3 = \dots = C_n
     \]

The final answer is \( \boxed{C_1 = 0 \text{ or } C_1 = 2} \)