Fix an integer $n\geq4$. Let $C_n$ be the collection of all $n$–point configurations in the plane, every three points of which span a triangle of area strictly greater than $1.$ For each configuration $C\in C_n$ let $f(n,C)$ be the maximal size of a subconfiguration of $C$ subject to the condition that every pair of distinct points has distance strictly greater than $2.$ Determine the minimum value $f(n)$ which $f(n,C)$ achieves as $C$ runs through $C_n.$

[i]Radu Bumbăcea and Călin Popescu[/i]
1. **Define the problem and notation:**
   - Fix an integer \( n \geq 4 \).
   - Let \( C_n \) be the collection of all \( n \)-point configurations in the plane, where every three points span a triangle of area strictly greater than 1.
   - For each configuration \( C \in C_n \), let \( f(n, C) \) be the maximal size of a subconfiguration of \( C \) such that every pair of distinct points has a distance strictly greater than 2.
   - Determine the minimum value \( f(n) \) which \( f(n, C) \) achieves as \( C \) runs through \( C_n \).

2. **Propose the minimum value:**
   - The required minimum is \( f(n) = \lceil n/3 \rceil \).

3. **Construct a configuration to achieve \( f(n) = \lceil n/3 \rceil \):**
   - Let \( m = \lceil n/3 \rceil \).
   - Consider the vertex set of \( m+1 \) equilateral triangles, \( \Delta_1, \ldots, \Delta_{m+1} \), each with side length 2.
   - These triangles are translated copies of one another, widely and suitably spread in the plane such that \( \text{dist}(\Delta_i, \Delta_j) > 2 \) for all \( i \neq j \).

4. **Remove vertices to form a configuration \( C \in C_n \):**
   - Remove \( 3m - n + 3 \) vertices from \( \Delta_{m+1} \).
   - This leaves a vertex configuration \( C \) in \( C_n \).

5. **Form a subconfiguration of size \( \lceil n/3 \rceil \):**
   - Choose one vertex from each \( \Delta_i \) (including \( \Delta_{m+1} \) if \( m < n/3 \)).
   - This subconfiguration has size \( \lceil n/3 \rceil \) and every two points are strictly more than 2 distance apart.

6. **Show that \( f(n, C) \leq \lceil n/3 \rceil \):**
   - Any subconfiguration of \( C \) larger than \( \lceil n/3 \rceil \) contains points 2 distance apart.
   - Thus, \( f(n, C) \leq \lceil n/3 \rceil \).

7. **Show that \( f(n) \geq \lceil n/3 \rceil \):**
   - It suffices to prove \( f(n, C) \geq \lceil n/3 \rceil \) for any configuration \( C \in C_n \).

8. **Consider the short segment graph \( G \) associated with \( C \):**
   - Points of \( C \) form the vertex set.
   - Short segments (length at most 2) in \( C \) form the edge set.

9. **Prove that \( G \) is 3-colorable:**
   - Induct on the number of vertices.
   - Base case: \( n = 3 \) is trivially 3-colorable.
   - Induction step: Let \( a \) be a vertex of \( G \).
     - If \( \text{deg } a \leq 2 \), a color is available for \( a \) to extend a valid 3-coloring of \( G - a \) to \( G \).
     - If \( \text{deg } a \geq 3 \), \( a \) has a neighbor \( b \) such that the line through \( a \) and \( b \) leaves neighbors of \( a \) on either side.
     - Let \( H \) and \( K \) be the short segment graphs associated with the subconfigurations on either side of the line.
     - By induction, \( H \) and \( K \) are 3-colorable.
     - Combine the 3-colorings of \( H \) and \( K \) to obtain a 3-coloring of \( G \).

10. **Conclude the proof:**
    - The most frequently colored points form a subconfiguration of at least \( n/3 \) points, each pair of which is strictly more than 2 distance apart.
    - Thus, \( f(n, C) \geq n/3 \).

\(\blacksquare\)

The final answer is \( \boxed{ \lceil n/3 \rceil } \)