AMC 12 Problem Sets¶

Here are 12 problems typical of AMC 12 competitions, which are aimed at high school students up to 12th grade. The difficulty levels range from basic algebra and geometry to more advanced topics like number theory and combinatorics:

1. Problem: Logarithms¶

Solve for \( x \) if \( \log_2 (x^2 - 5x + 6) = 2 \).

2. Problem: Area of a Triangle¶

The base of a triangle is increased by 50%, and its height is decreased by 20%. By what percentage does the area of the triangle change?

3. Problem: Polynomial Roots¶

Find the sum of the squares of the roots of the equation \( x^2 - 5x + 6 = 0 \).

4. Problem: Probability¶

A box contains 5 red marbles, 3 green marbles, and 2 blue marbles. If two marbles are drawn at random, what is the probability that both marbles are the same color?

5. Problem: Number Theory¶

Find the greatest common divisor (gcd) of \( 48 \) and \( 180 \).

6. Problem: Counting and Probability¶

How many ways can the letters in the word MISSISSIPPI be rearranged?

7. Problem: Geometry and Circles¶

In a circle, a chord of length \( 10 \) is 6 units away from the center. What is the radius of the circle?

8. Problem: Arithmetic Sequence¶

The sum of the first 10 terms of an arithmetic sequence is \( 100 \), and the sum of the first 20 terms is \( 300 \). What is the first term?

9. Problem: Sum of a Series¶

Find the sum of the infinite geometric series \( 5 + 4 + 3.2 + \dots \).

10. Problem: Complex Numbers¶

What is the value of \( i^7 \), where \( i \) is the imaginary unit?

11. Problem: Trigonometry¶

If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is in the first quadrant, find \( \cos \theta \).

12. Problem: Vectors¶

Let \( \mathbf{u} = \langle 1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 4 \rangle \). Find \( \mathbf{u} \cdot \mathbf{v} \).

These are sample AMC 12-style problems covering a variety of mathematical concepts.

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