Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.
1. **Define the sets and their properties:**
   Let \( A_1, A_2, \ldots, A_n \) be the sets representing the students who got the best scores in each of the \( n \) subjects. Each set \( A_k \) has exactly 3 elements, i.e., \( |A_k| = 3 \) for all \( k \). Additionally, for any two distinct subjects \( i \) and \( j \), the intersection \( A_i \cap A_j \) contains exactly one element, i.e., \( |A_i \cap A_j| = 1 \) for all \( i \neq j \).

2. **Assume a student appears in the most number of sets:**
   Let \( 1 \) be the student who appears in the most number of sets. Without loss of generality, assume \( 1 \in A_1, A_2, \ldots, A_m \) and \( 1 \notin A_k \) for \( k > m \). We need to determine the maximum value of \( m \).

3. **Analyze the intersections:**
   If \( 1 \) appears in \( m \) sets, then for each \( j > m \), the sets \( A_j \cap A_k \) (for \( k = 1, 2, \ldots, m \)) must be distinct elements other than \( 1 \). Since \( |A_j| = 3 \), \( A_j \) can have at most 2 other elements besides \( 1 \). Therefore, \( m \leq 3 \).

4. **Consider the remaining sets:**
   For \( j, k > m \), the intersections \( A_i \cap A_j \) and \( A_i \cap A_k \) must be distinct for different \( i \). Since each set \( A_i \) has exactly 3 elements, and each element can appear in at most 3 sets, the number of sets \( A_j \) with \( j > m \) is limited.

5. **Calculate the maximum number of sets:**
   Since \( |A_1 \setminus \{1\}| = 2 \), there can be at most \( 2 \times 2 = 4 \) sets \( A_j \) with \( j > m \). Therefore, the total number of sets \( n \) is at most \( m + 4 \). Given \( m \leq 3 \), we have \( n \leq 7 \).

6. **Construct an example for \( n = 7 \):**
   Consider the sets:
   \[
   \begin{aligned}
   &A_1 = \{1, 2, 3\}, \\
   &A_2 = \{1, 4, 5\}, \\
   &A_3 = \{1, 6, 7\}, \\
   &A_4 = \{2, 4, 6\}, \\
   &A_5 = \{2, 5, 7\}, \\
   &A_6 = \{3, 4, 7\}, \\
   &A_7 = \{3, 5, 6\}.
   \end{aligned}
   \]
   These sets satisfy the given conditions: each set has 3 elements, and the intersection of any two sets has exactly one element.

7. **Conclusion:**
   If \( n \geq 8 \), then one student must have received the top mark in all subjects, as it would be impossible to satisfy the given conditions otherwise.

The final answer is \( \boxed{ n = 7 } \).