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This is a set of multiple-choice math questions for middle school Olympiad preparation. Let's go through each one systematically:

### 1. The arrangement of tables and chairs in a rectangular classroom. Based on the seating described, find the total number of students.
- This is a logic puzzle where the seating arrangement follows specific positions. The full classroom needs to be calculated.
  
### 2. Arrangement of three large exponential numbers \( 2^{4444} \cdot 3^{5555}, 3^{3333} \cdot 5^{2222}, 5^{2222} \cdot 2^{4444} \).
- The question asks for ordering these from smallest to largest by comparing their sizes.

### 3. Six congruent circles fit inside a square, with the circumference of each circle being 62.8 cm. The task is to calculate the shaded area.
- Use the given circumference to determine the radius, then calculate the area of the square minus the combined area of the circles.

### 4. Five goats consume grass on 5 acres in 5 days. How long will 3 goats take to consume grass on 3 acres?
- This is a classic problem in rates and proportions.

### 5. A person travels 70 km in 2.5 hours. To arrive 3/4 hour earlier, what must be their new average speed?
- You need to calculate the time saved and the new speed required.

### 6. Find the last digit of the sum \( 1 + 2 + 3 + \ldots + 2017 \).
- This is an arithmetic series where you need the last digit of the sum.

### 7. If \( y \) and \( z \) are integers, \( y \geq z \), and the equation involves square roots and sums, find \( y - z \).

### 8. Solve the equation involving powers of \( p \) and \( q \), to determine the value of \( x \).

### 9. If adding 326 to the number \( 2A3 \) results in a number divisible by 9, what is \( A + B \)?

### 10. If the difference between two numbers is 2, and the square of their difference is 6, find their sum.
- Solve for the two unknowns using algebraic methods.

### 11. If \( a \) and \( b \) satisfy the equation \( 3^{a+1} - 3 \cdot 2^b = -3 \) and \( 2 \cdot 3^a + 2^b = 10 \), find \( a \times b \).

If you'd like detailed solutions for any specific question, feel free to ask!

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Here are some relevant follow-up questions:

1. How do you compare large exponential numbers like in question 2?
2. Can you explain how to approach proportional rate problems, such as question 4?
3. What is the method to calculate areas with inscribed circles (like in question 3)?
4. How do you handle arithmetic series and find the last digit of large sums?
5. What is the best strategy for solving systems of equations with exponents, such as in question 11?

**Tip:** For complex problems involving exponentiation and large numbers, break the problem into parts using logarithmic approximations for easier comparison.

This is a set of multiple-choice questions for middle school Olympiad math preparation, including topics like seating arrangements, large exponential numbers, circle area calculations, proportional rates, arithmetic series, and algebraic equations.

Explore a series of middle school math Olympiad problems covering geometry, exponential number comparison, area calculations, arithmetic series, and algebraic equations. Detailed solutions and explanations for each problem are provided.

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