Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$.

[i]Proposed by Alexander Ivanov, Bulgaria[/i]
1. **Define the Problem and Setup:**
   We are given an \( n \)-gon with \( n-3 \) black diagonals and \( n-3 \) red diagonals such that no two diagonals of the same color intersect strictly inside the polygon. We need to find the maximum number of intersection points between diagonals of different colors strictly inside the polygon.

2. **Define the Length of a Chord:**
   Consider a chord \( XY \) in the \( n \)-gon. Let \( a \) points be on one side of the chord and \( b \) points on the other side, so \( a + b = n - 2 \). Define the length \( \ell(XY) \) of the chord \( XY \) as \( \min\{a, b\} \).

3. **Lemma:**
   Suppose we have two non-intersecting chords \( AB \) and \( CD \) of lengths \( x \) and \( y \). For any triangulation of the \( n \)-gon, the number of interior intersections with these two chords is bounded above by \( x + y + n - 4 \).

4. **Proof of Lemma:**
   - Label the points starting from \( A \) as \( A, P_1, \ldots, P_x, B, U_1, \ldots, U_t, C, Q_1, \ldots, Q_y, D, V_1, \ldots, V_s \), where \( x + y + t + s = n - 4 \).
   - Define \( U \)-good and \( V \)-good points based on whether they connect to \( U_j \) or \( V_j \) respectively.
   - Count the number of intersections of edges from \( P_i \) or \( Q_i \) to \( U_j \) with \( AB \cup CD \). Let \( P_{a_1}, \ldots, P_{a_g} \) be the \( U \)-good \( P_i \)'s and \( Q_{b_1}, \ldots, Q_{b_h} \) be the \( U \)-good \( Q_i \)'s.
   - The intervals \( [\min S_{P_i}, \max S_{P_i}] \) and \( [\min S_{Q_i}, \max S_{Q_i}] \) do not intersect in their interiors.
   - The number of intersections of edges from \( P_i \) or \( Q_i \) to \( U_j \) with \( AB \cup CD \) is bounded above by \( t - 1 \) plus the number of \( U \)-good points.
   - Similarly, the number of intersections of edges from \( P_i \) or \( Q_i \) to \( V_j \) with \( AB \cup CD \) is bounded above by \( s - 1 \) plus the number of \( V \)-good points.
   - Count the number of edges between \( P_i \)'s and \( Q_j \)'s. Suppose there are \( g_u \) \( U \)-good \( P_i \)'s and \( g_v \) \( V \)-good \( P_i \)'s.
   - The number of possible edges is at most \( (x + 2 - g_u - g_v) + (y + 2 - g_u' - g_v') - 1 \).
   - The total number of intersections is at most \( 2(x + y) + t + s + 4 - (g_u + g_v + g_u' + g_v') \leq 2(x + y) + t + s = n - 4 + x + y \).

5. **Summing the Bounds:**
   - For odd \( n \), let \( d_1 \leq d_2 \leq \cdots \leq d_{n-3} \) be the lengths of the chords of the red triangulation. The total red-black intersections are bounded by \( (d_1 + \cdots + d_{n-3}) + \frac{n-3}{2}(n-4) \).
   - By induction, \( d_1, d_2 \leq 1 \), \( d_3, d_4 \leq 2 \), and so on until \( d_{n-4}, d_{n-3} \leq \frac{n-3}{2} \).
   - The total number of intersections is at most \( \frac{n-3}{2} \cdot \frac{n-1}{2} + (n-4) \frac{n-3}{2} = \frac{3(n-3)^2}{4} \).

6. **Even \( n \):**
   - The number of red-black intersections with any given red chord is at most \( n-3 \).
   - The number of red-black intersections is bounded by \( \left(1 + 1 + 2 + 2 + \cdots + \frac{n-4}{2} + \frac{n-4}{2}\right) + \frac{n-4}{2}(n-4) + n-3 = \frac{3(n-3)^2 + 1}{4} \).

7. **Conclusion:**
   - The maximum number of intersection points between diagonals of different colors is \( \boxed{\lceil \frac{3}{4}(n-3)^2 \rceil} \).