More than five competitors participated in a chess tournament. Each competitor played exactly once against each of the other competitors. Five of the competitors they each lost exactly two games. All other competitors each won exactly three games. There were no draws in the tournament. Determine how many competitors there were and show a tournament that verifies all conditions.
1. **Define the number of competitors:**
   Let the number of competitors in the tournament be \( n \).

2. **Analyze the first five competitors:**
   Each of these five competitors lost exactly 2 games. Therefore, each of these five competitors won \( n - 1 - 2 = n - 3 \) games.
   - Total games lost by these five competitors: \( 5 \times 2 = 10 \)
   - Total games won by these five competitors: \( 5 \times (n - 3) = 5(n - 3) \)

3. **Analyze the remaining competitors:**
   Each of the remaining \( n - 5 \) competitors won exactly 3 games. Therefore, each of these competitors lost \( n - 1 - 3 = n - 4 \) games.
   - Total games won by these remaining competitors: \( 3 \times (n - 5) = 3(n - 5) \)
   - Total games lost by these remaining competitors: \( (n - 5) \times (n - 4) = (n - 5)(n - 4) \)

4. **Set up the equation for total games won and lost:**
   Since the total number of games won must equal the total number of games lost, we have:
   \[
   5(n - 3) + 3(n - 5) = 10 + (n - 5)(n - 4)
   \]

5. **Simplify the equation:**
   \[
   5(n - 3) + 3(n - 5) = 10 + (n - 5)(n - 4)
   \]
   \[
   5n - 15 + 3n - 15 = 10 + (n^2 - 9n + 20)
   \]
   \[
   8n - 30 = 10 + n^2 - 9n + 20
   \]
   \[
   8n - 30 = n^2 - 9n + 30
   \]
   \[
   0 = n^2 - 17n + 60
   \]

6. **Solve the quadratic equation:**
   \[
   n^2 - 17n + 60 = 0
   \]
   Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -17 \), and \( c = 60 \):
   \[
   n = \frac{17 \pm \sqrt{289 - 240}}{2}
   \]
   \[
   n = \frac{17 \pm \sqrt{49}}{2}
   \]
   \[
   n = \frac{17 \pm 7}{2}
   \]
   \[
   n = \frac{24}{2} = 12 \quad \text{or} \quad n = \frac{10}{2} = 5
   \]

7. **Verify the solution:**
   Since \( n > 5 \), we discard \( n = 5 \) and accept \( n = 12 \).

8. **Construct a tournament that verifies all conditions:**
   - Five competitors each lost exactly 2 games and won \( 12 - 3 = 9 \) games.
   - The remaining 7 competitors each won exactly 3 games and lost \( 12 - 4 = 8 \) games.

   This setup satisfies all the given conditions.

The final answer is \( \boxed{ n = 12 } \)