Ravi and eight of his colleagues took a 90-point assessment. Each of them received a unique integer score, and their average score was 80. Ravi scored 83, and his score was higher than exactly two of his colleagues. What could be the maximum possible absolute difference between the scores of two of his colleagues?"

Given that there are 9 people who each took a 90-point assessment, we can approach this problem step-by-step as follows:

1. **Calculate the Total Score**:  
   Since the average score is 80 for all nine participants, their total score is:
   \[
   9 \times 80 = 720
   \]

2. **Identify Ravi's Score**:  
   Ravi scored 83. Since his score is higher than exactly two of his colleagues, this means he has the 7th highest score in ascending order, with only two participants scoring lower than him.

3. **Assign Unique Integer Scores**:  
   Let's assign scores in ascending order, keeping Ravi at the 7th position with a score of 83:
   \[
   a_1 < a_2 < \ldots < a_9
   \]
   where \( a_7 = 83 \).

4. **Find Possible Scores for Each Person**:
   To maximize the absolute difference between the highest and lowest scores, we will:
   - Set scores for \(a_1\) and \(a_2\) to be as low as possible.
   - Set scores for \(a_8\) and \(a_9\) to be as high as possible.

5. **Create an Equation for the Scores**:
   Let’s define:
   \[
   a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + 83 + a_8 + a_9 = 720
   \]
   This simplifies to:
   \[
   a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_8 + a_9 = 637
   \]

6. **Determine Maximal Absolute Difference**:
   By strategically choosing low values for \( a_1 \) and \( a_2 \) and high values for \( a_8 \) and \( a_9 \), we maximize \( a_9 - a_1 \).

After solving through the possible values satisfying these conditions, the **maximum possible absolute difference** between the highest and lowest scores among Ravi’s colleagues is found to be **44**.

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Would you like further details on this solution? 

Here are some related questions for further practice:
1. If Ravi scored the median score, what would his score be?
2. What is the minimum possible score one of Ravi's colleagues could receive?
3. What is the minimum possible absolute difference between the scores of two of his colleagues?
4. How would the average change if Ravi’s score were 5 points higher?
5. Can the maximum absolute difference be higher if the average score is different?

**Tip:** To maximize a range with constraints, try minimizing the smallest value and maximizing the largest within the given conditions.

Ravi and eight of his colleagues took a 90-point assessment. Each of them received a unique integer score, and their average score was 80. Ravi scored 83, and his score was higher than exactly two of his colleagues. What could be the maximum possible absolute difference between the scores of two of his colleagues?

Explore how to find the maximum possible absolute difference between the scores of nine colleagues who took a 90-point assessment, given an average score of 80 and unique integer scores. Detailed solution with step-by-step optimization approach suitable for Grades 10-12.

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