The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled?

[i]Proposed by A. Golovanov[/i]
1. **Understanding the Problem:**
   We are given a regular 2012-gon with vertices labeled \( A_1, A_2, \ldots, A_{2012} \). The problem states that if \( k + \ell \) and \( m + n \) leave the same remainder when divided by 2012, then the chords \( A_kA_\ell \) and \( A_mA_n \) do not intersect. We know the first two vertices are labeled \( A_1 \) and \( A_4 \). We need to determine the label of the tenth vertex.

2. **Analyzing the Condition:**
   Consider three consecutive vertices labeled \( A_k, A_m, A_\ell \). We need to find a unique \( 1 \leq n \leq 2012 \) such that \( n \equiv (k + \ell) - m \pmod{2012} \). Since the chords \( A_kA_\ell \) and \( A_mA_n \) would intersect if \( n \neq m \), the only possibility is \( n = m \). This forces \( k + \ell \) to be even, and \( \ell \equiv 2m - k \pmod{2012} \).

3. **Interpreting the Labeling:**
   This condition implies that the labels follow an arithmetic progression with a common difference. Given \( k = 1 \) and \( m = 4 \), the common difference is \( 4 - 1 = 3 \). Therefore, the \( t \)-th vertex is labeled \( A_{1 + 3(t - 1) \pmod{2012}} \).

4. **Finding the Tenth Vertex:**
   To find the label of the tenth vertex, we substitute \( t = 10 \) into the formula:
   \[
   1 + 3(10 - 1) \pmod{2012} = 1 + 3 \times 9 \pmod{2012} = 1 + 27 \pmod{2012} = 28
   \]
   Thus, the tenth vertex is labeled \( A_{28} \).

5. **Verification:**
   We need to ensure that this labeling satisfies the given condition. Since \( 3 \nmid 2012 \), the values \( 1 + 3(t - 1) \pmod{2012} \) cover all indices \( \{1, 2, \ldots, 2012\} \). For two vertices labeled \( 1 + 3(k - 1) \pmod{2012} \) and \( 1 + 3(\ell - 1) \pmod{2012} \), the vertex labeled \( 1 + 3(m - 1) \pmod{2012} \) corresponds to \( n = (k + \ell) - m \), ensuring \( k < n < \ell \) and the chords do not intersect. Therefore, the labeling is consistent with the problem's conditions.

The final answer is \( \boxed{A_{28}} \).