The uploaded document is a multiple-choice math competition paper, including questions on various mathematical topics such as algebra, geometry, and number theory. Here are the main topics and ideas covered, followed by a set of sample questions that align with the style of the original paper. I've also added diagrams to complement the questions.

Main Topics Covered:

1. Work and Time Problems: Related to combined rates of work.
2. Consecutive Integers: Sum and product of consecutive numbers.
3. Quadratic Equations: Properties of the roots.
4. Geometry: Area of triangles, grids, and diagrams.
5. Number Theory: Special types of numbers and their properties.

Sample Problems with Diagrams:

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1. Work and Time (Rate Problems)
Question: Alex can paint a wall in 45 minutes, and Ben can paint the same wall in 30 minutes. If they work together, how many walls can they paint in 2 hours?
A) 3 walls
B) 4 walls
C) 5 walls
D) 6 walls
E) 7 walls

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2. Geometry - Triangle Midpoints (Diagram Provided Below)
Question: In triangle $ABC$, $M$, $N$, and $P$ are the midpoints of sides $BC$, $CA$, and $AB$ respectively. If the area of triangle $ABC$ is 16, what is the area of triangle $MNP$?
A) 2
B) 4
C) 6
D) 8
E) 10
(Diagram: A triangle with points M, N, and P marked as midpoints on the sides.)

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3. Grid and Diagonal (Diagram Provided Below)
Question: A $5 \times 7$ grid of squares has a diagonal drawn from the top-left corner to the bottom-right corner. How many squares does the diagonal pass through?
A) 7
B) 9
C) 11
D) 13
E) 15
(Diagram: A grid with a diagonal line passing through multiple squares.)

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4. Number Theory - Prime Expressions
Question: Choose two positive integers $m$ and $n$ such that $m > n$, and as many of the following expressions are prime: $m+n$, $m-n$, $m \times n$, $m \div n$. How many of these expressions are prime?
A) 0
B) 1
C) 2
D) 3
E) 4

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5. Geometry - Semicircle and Circle (Diagram Provided Below)
Question: A circle of radius 3 is inscribed in a semicircle with radius 6. What fraction of the area of the semicircle is outside the circle (shaded region)?
A) $\frac{1}{2}$
B) $\frac{\pi}{3}$
C) $\frac{2}{\pi}$
D) $\frac{3}{4}$
E) $\frac{\pi - 1}{\pi}$
(Diagram: A semicircle with a smaller inscribed circle, showing the shaded area between the two.)

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These problems cover a range of mathematical skills and problem-solving techniques. If you need detailed diagrams or more questions, feel free to ask.

5 Relative Questions:

1. Can you explain how to solve problems involving the rates of two or more workers?
2. How do you calculate the area of triangles formed by midpoints in geometry?
3. How do diagonals interact with grids in terms of counting the squares they pass through?
4. What are some strategies for determining prime numbers in algebraic expressions?
5. How can geometric shapes, like semicircles and circles, be used to form shaded area problems?

Tip:

When working on grid problems with diagonals, a good technique is to apply the greatest common divisor (GCD) of the grid dimensions to determine how many squares a diagonal passes through.