https://file.aiquickdraw.com/mathbot/file/17268938787074mlwogdh.pdf

can you create a similar paper with a little more difficulty paper on the same topics

I can help create a more difficult version of the exam paper based on the current topics. The paper provided covers various mathematical areas, such as:

- Arithmetic (work problems, sum of integers)
- Algebra (quadratics, systems of equations)
- Geometry (areas of triangles, midpoints, semicircles)
- Combinatorics and number theory (hope numbers, expressions involving prime numbers)
- Logarithms, roots, and properties of numbers

To make the paper more challenging, I will adjust by:

1. Including more advanced algebraic manipulations (e.g., involving higher-degree polynomials).
2. Adding tougher geometry problems, including proofs or additional constraints.
3. Introducing number theory concepts like modular arithmetic or Diophantine equations.
4. Extending some combinatorial problems to require deeper reasoning or additional steps.
5. Incorporating more advanced calculus topics, if applicable.

Here’s a sample of how I might adjust some questions for greater difficulty:

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### 1. **(Work Rate Problem - Harder Version)**
Abey can mow a lawn in 30 minutes, and Rose can mow a lawn in 15 minutes. However, every 10 minutes, Abey needs to take a 2-minute break, and Rose needs to take a 1-minute break. How many lawns can they mow together in 1 hour under these conditions?

A) 2 B) 3 C) 4 D) 5 E) 6

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### 2. **(Algebra - Quadratics with Complex Roots)**
If \( a \) and \( b \) are the roots of the equation \( x^2 - 7x + 11 = 0 \), find \( a^3 + b^3 \).

A) 25 B) 34 C) 61 D) 89 E) 133

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### 3. **(Geometry - Extended Triangle Problem)**
In triangle \( ABC \), points \( M \), \( N \), and \( P \) are the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. The area of triangle \( ABC \) is 24, and the line connecting \( M \) to \( N \) is extended to intersect the circumcircle of \( ABC \) again at point \( D \). Find the area of triangle \( MND \).

A) 4 B) 6 C) 8 D) 10 E) 12

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### 4. **(Number Theory - Advanced Modular Arithmetic)**
Let \( p \) be a prime number and \( a, b \) be integers such that \( a^2 \equiv b^2 \pmod{p} \). If \( a \neq b \), find the number of solutions to the equation for \( p = 17 \).

A) 8 B) 9 C) 10 D) 12 E) 16

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### 5. **(Combinatorics - Extended Sequence Problem)**
You are given a sequence of shapes, each composed of small shaded squares. The \( n \)-th shape consists of \( 3n^2 + 2n \) squares. Find the number of shaded squares in the 20th shape.

A) 1280 B) 1320 C) 1440 D) 1600 E) 1800

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Would you like me to expand this into a complete paper? Feel free to request adjustments or specify particular topics you’d like to emphasize.

### Relative Questions:
1. How would you approach work rate problems with breaks?
2. What methods can be used to calculate sums of cubes of quadratic roots?
3. How does the extension of a line affect triangle geometry in advanced problems?
4. What are the strategies for solving modular arithmetic equations involving primes?
5. How does a sequence’s growth impact combinatorial counting in advanced shapes?

**Tip:** For more challenging geometry problems, always review the properties of midpoints and circumcircles!

This math challenge paper for Grades 9-11 covers key topics like algebra, geometry, combinatorics, and number theory. Designed to push your problem-solving skills, this paper includes harder versions of questions involving work rates, quadratic roots, triangle areas, and prime numbers.

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