Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle.

Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A [i]cool procedure[/i] consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of the lamp; if $i$ is turned on, a [i]signal[/i] of range $i$ is sent by this lamp, and it will be received only by the next $i$ lamps which follow $i$, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed.

The example in attachment, for $n=4$, ilustrates a configuration where lamps $2$ and $4$  are initially turned on. Lamp $2$ sends signal only for the lamps $3$ e $4$, while lamp $4$ sends signal for lamps $1$, $2$, $3$ e $4$. Therefore, we verify that lamps $1$ e $2$ received only one signal, while lamps $3$ e $4$ received two signals. Therefore, in the next configuration, lamps $1$ e $4$ will be turned on, while lamps $2$ e $3$ will be turned off.

Let $\Psi$ to be the set of all $2^n$ possible configurations, where $0 \le \ell \le n$ random lamps are turned on. We define a function $f: \Psi \rightarrow \Psi$ where, if $\xi$ is a configuration of lamps, then $f(\xi)$ is the configurations obtained after we perform the [i]cool procedure[/i] described above.

Determine all values of $n$ for which $f$ is bijective.
To determine all values of \( n \) for which the function \( f \) is bijective, we need to analyze the behavior of the function \( f \) defined on the set of all possible configurations of \( n \) lamps. We will use polynomial theory and properties of finite fields to achieve this.

1. **Polynomial Representation**:
   Consider the configuration of lamps as a polynomial \( g_a(x) \) in \( \mathbb{F}_2[x] \) modulo \( x^n - 1 \). Each lamp configuration can be represented as a polynomial where the coefficient of \( x^i \) is 1 if the \( i \)-th lamp is on and 0 if it is off.

2. **Function Definition**:
   The function \( f \) can be expressed in terms of polynomials as:
   \[
   f(g_a) = \frac{x g_a(x^2) - g_a(x)}{x-1} \pmod{x^n - 1}
   \]
   This function describes the transformation of the lamp configuration after performing the cool procedure.

3. **Bijectivity Condition**:
   For \( f \) to be bijective, it must be injective. This means that if \( f(g_a) = 0 \), then \( g_a \) must be 0. We need to analyze when \( f(g_a) = 0 \) implies \( g_a = 0 \).

4. **Factorization of \( n \)**:
   Let \( n = 2^k \cdot m \), where \( m \) is odd. We will consider two cases: \( m = 1 \) and \( m > 1 \).

5. **Case \( m > 1 \)**:
   - **Lemma**: If \( x^n - 1 \mid x g_a(x^2) - g_a(x) \), then:
     \[
     \frac{x g_a(x^2) - g_a(x)}{x-1} = 0 \quad \text{or} \quad \frac{x^n - 1}{x-1} \pmod{x^n - 1}
     \]
   - **Proof of Lemma**: Since \( \frac{x^n - 1}{x-1} \) divides \( \frac{x g_a(x^2) - g_a(x)}{x-1} \), we have:
     \[
     \frac{x g_a(x^2) - g_a(x)}{x-1} = q(x) \cdot \frac{x^n - 1}{x-1}
     \]
     for some polynomial \( q(x) \). This implies that the lemma holds.

   - **Permutation Analysis**: Consider the function \( h(x) = 2x + 1 \pmod{n} \). This function is a permutation on the set \( \{0, 1, \ldots, n-1\} \) with at least two cycles when \( m > 1 \). This is because \( h(m \cdot 2^k - 1) = m \cdot 2^k - 1 \), forming a cycle of length 1.

   - **Cycle Construction**: For a cycle of length \( j \) containing elements \( c_1, c_2, \ldots, c_j \), construct the polynomial:
     \[
     g_a(x) = \sum_{i=1}^j x^{c_i}
     \]
     This polynomial satisfies \( x^n - 1 \mid x g_a(x^2) - g_a(x) \). Since there are at least two cycles, either one of them is 0, or the sum of them will be 0. Thus, there exists a non-zero polynomial \( g \) such that \( f(g) = 0 \).

6. **Case \( m = 1 \)**:
   - When \( n = 2^k \), we need to show that \( f \) is bijective. Consider the function \( h(x) = 2x + 1 \pmod{n} \). If \( g_a(x) \) has a term \( x^m \) where \( m \neq 2^k - 1 \), then:
     \[
     x^n - 1 \mid x g_a(x^2) - g_a(x)
     \]
     By analyzing the \( 2 \)-adic valuation \( \upsilon_2 \), we can show that no such \( m_1 \) exists, leading to a contradiction. Therefore, \( g_a(x) = x^{2^k - 1} \) gives \( f(g_a) = \frac{x^{2^k} - 1}{x-1} \neq 0 \).

Conclusion:
- For \( n = 2^k \), the function \( f \) is bijective.
- For \( n = 2^k \cdot m \) with \( m > 1 \), the function \( f \) is not bijective.

The final answer is \( \boxed{ n = 2^k } \).