We wish to write $n$ distinct real numbers $(n\geq3)$ on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of $n$ such that this is possible.
1. Let \( x_i \) be a number on the circumference of the circle. Define \( x_{i+1} \) as the number directly next to \( x_i \) in the clockwise direction and \( x_{i-1} \) as the number directly next to \( x_i \) in the counterclockwise direction. According to the problem, each number is equal to the product of its immediate neighbors:
   \[
   x_i = x_{i-1} x_{i+1}
   \]

2. Similarly, for \( x_{i-1} \) and \( x_{i+1} \), we have:
   \[
   x_{i-1} = x_{i-2} x_i
   \]
   \[
   x_{i+1} = x_i x_{i+2}
   \]

3. Substitute \( x_{i+1} \) from the second equation into the first equation:
   \[
   x_i = x_{i-1} (x_i x_{i+2})
   \]
   Simplifying, we get:
   \[
   x_i = x_{i-1} x_i x_{i+2}
   \]
   Dividing both sides by \( x_i \) (assuming \( x_i \neq 0 \)):
   \[
   1 = x_{i-1} x_{i+2}
   \]
   Therefore:
   \[
   x_{i-1} x_{i+2} = 1
   \]

4. Similarly, substitute \( x_{i-1} \) from the first equation into the second equation:
   \[
   x_{i-1} = x_{i-2} (x_{i-1} x_{i+1})
   \]
   Simplifying, we get:
   \[
   x_{i-1} = x_{i-2} x_{i-1} x_{i+1}
   \]
   Dividing both sides by \( x_{i-1} \) (assuming \( x_{i-1} \neq 0 \)):
   \[
   1 = x_{i-2} x_{i+1}
   \]
   Therefore:
   \[
   x_{i-2} x_{i+1} = 1
   \]

5. From the above, we have:
   \[
   x_{i-2} x_{i+1} = 1 \quad \text{and} \quad x_{i-1} x_{i+2} = 1
   \]

6. This implies that for any \( x_k \) in the circumference:
   \[
   x_k x_{k+3} = 1 \quad \text{and} \quad x_k x_{k-3} = 1
   \]

7. Notice that \( x_{k-3} = x_{k+3} \), which would break the conditions set by the problem unless \( n = 6 \) because going 3 units in the clockwise direction would take us to the same place as going 3 units in the counterclockwise direction.

8. Therefore, the only possible value for \( n \) is 6. When \( n = 6 \), we have the solution:
   \[
   x_1, x_1 x_2, x_2, \frac{1}{x_1}, \frac{1}{x_1 x_2}, \frac{1}{x_2}
   \]
   where \( x_1 \) and \( x_2 \) represent 2 distinct real numbers.

The final answer is \( \boxed{ n = 6 } \).