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This question presents a scoring system for a Mathematics Olympiad quiz. Here's a breakdown of the problem:

### Problem:
- **Shirley attempted all 30 questions**.
- For each **correct answer**, she earns **5 points**.
- For each **wrong answer or unanswered question**, **2 points** are deducted.
- Her total score for the quiz is **not more than 66**.

### We need to find the maximum number of correct answers she could have obtained.

### Solution:
Let:
- \( x \) = number of correct answers.
- \( y \) = number of wrong answers (since she attempted all 30 questions, \( y = 30 - x \)).

#### Total score equation:
\[
\text{Score} = 5x - 2y
\]
Substitute \( y = 30 - x \) into the equation:
\[
\text{Score} = 5x - 2(30 - x)
\]
Simplifying:
\[
\text{Score} = 5x - 60 + 2x = 7x - 60
\]
We are told the score is not more than 66, so:
\[
7x - 60 \leq 66
\]
Add 60 to both sides:
\[
7x \leq 126
\]
Divide by 7:
\[
x \leq 18
\]
Thus, Shirley can have a **maximum of 18 correct answers**.

### Final Answer:
The maximum number of correct answers Shirley obtained is **18**.

Would you like more details or have any questions?

Here are 5 related questions for further understanding:
1. What would the equation be if 3 points were deducted for each wrong answer instead of 2?
2. How would the solution change if Shirley had attempted only 25 questions?
3. Can you generalize the approach to find the maximum score if the number of questions was different?
4. What if Shirley's score was not more than 70 instead of 66?
5. What is the minimum number of correct answers she could have obtained while still attempting all 30 questions?

### Tip:
When solving word problems involving systems like scoring rules, always break down the rules into mathematical expressions step by step.

In a Mathematics Olympiad quiz, 5 points are awarded for a correct answer and 2 points are deducted for a wrong answer or if a question is left unanswered. Shirley attempted all 30 questions and her total score for the quiz was not more than 66. Find the maximum number of correct answers she obtained.

This problem involves finding the maximum number of correct answers in a 30-question Mathematics Olympiad quiz. Each correct answer earns 5 points, while 2 points are deducted for wrong or unanswered questions. The total score is limited to no more than 66 points. Explore the algebraic approach and inequality solving to determine the solution.

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