Let $ p \geq 3$ be a prime, and let $ p$ points $ A_{0}, \ldots, A_{p-1}$ lie on a circle in that order. Above the point $ A_{1+\cdots+k-1}$ we write the number $ k$ for $ k=1, \ldots, p$ (so $ 1$ is written above $ A_{0}$). How many points have at least one number written above them?
1. **Define the function and the problem setup:**
   Let \( p \geq 3 \) be a prime number, and consider \( p \) points \( A_0, A_1, \ldots, A_{p-1} \) arranged in a circle. We write the number \( k \) above the point \( A_{0+1+\cdots+(k-1)} \) for \( k = 1, 2, \ldots, p \). This means that the number \( k \) is written above the point \( A_{\frac{k(k-1)}{2} \mod p} \).

2. **Define the function \( f \):**
   Define the function \( f: \{1, 2, \ldots, p\} \to \{0, 1, \ldots, p-1\} \) by \( f(k) = \left(\frac{k(k-1)}{2}\right) \mod p \). This function maps each \( k \) to the point \( A_{\frac{k(k-1)}{2} \mod p} \).

3. **Check for distinctness:**
   We need to determine when \( f(a) = f(b) \) for distinct \( a, b \in \{1, 2, \ldots, p\} \). Suppose \( f(a) = f(b) \). Then:
   \[
   \frac{a(a-1)}{2} \equiv \frac{b(b-1)}{2} \pmod{p}
   \]
   Multiplying both sides by 2 (since \( p \) is an odd prime, 2 has an inverse modulo \( p \)):
   \[
   a(a-1) \equiv b(b-1) \pmod{p}
   \]
   Rearranging terms, we get:
   \[
   a^2 - a \equiv b^2 - b \pmod{p}
   \]
   \[
   a^2 - b^2 \equiv a - b \pmod{p}
   \]
   \[
   (a - b)(a + b - 1) \equiv 0 \pmod{p}
   \]
   Since \( p \) is a prime, this implies either \( a \equiv b \pmod{p} \) or \( a + b \equiv 1 \pmod{p} \).

4. **Analyze the solutions:**
   - If \( a \equiv b \pmod{p} \), then \( a = b \), which contradicts the assumption that \( a \) and \( b \) are distinct.
   - Therefore, \( a + b \equiv 1 \pmod{p} \).

5. **Count the distinct values:**
   For each \( k \) in \( \{1, 2, \ldots, p\} \), there is a unique \( k' \) such that \( k + k' \equiv 1 \pmod{p} \). This means that the values of \( f(k) \) are distinct for \( k \) in \( \{1, 2, \ldots, \frac{p+1}{2} \} \).

6. **Conclusion:**
   The number of distinct points \( A_i \) that have at least one number written above them is \( \frac{p+1}{2} \).

The final answer is \( \boxed{\frac{p+1}{2}} \)