AIME Problem Sets¶
Here are 12 problems typical of the AIME (American Invitational Mathematics Examination), which is known for its challenging, proof-based format. Each problem requires a deep understanding of mathematical concepts and typically involves algebra, geometry, number theory, or combinatorics.
1. Problem: Algebra and Polynomials¶
Let \( P(x) \) be a polynomial of degree 3 such that \( P(1) = 10 \), \( P(2) = 20 \), \( P(3) = 30 \), and \( P(4) = 40 \). Find \( P(5) \).
2. Problem: Number Theory¶
Find the smallest positive integer \( n \) such that \( n! \) is divisible by \( 2023 \).
3. Problem: Geometry¶
In triangle \( ABC \), \( AB = 13 \), \( AC = 14 \), and \( BC = 15 \). Find the length of the altitude from \( A \) to \( BC \).
4. Problem: Combinatorics¶
In how many ways can 5 indistinguishable apples be distributed among 3 distinguishable boxes such that no box is empty?
5. Problem: Trigonometry¶
Let \( \theta \) be an angle such that \( \cos(2\theta) = \frac{7}{9} \). Find the value of \( \sin(\theta) \).
6. Problem: Arithmetic Sequences¶
The arithmetic sequence \( a_1, a_2, a_3, \dots \) satisfies \( a_1 + a_2 + a_3 + \dots + a_{20} = 210 \) and \( a_1 + a_2 + a_3 + \dots + a_{30} = 465 \). Find \( a_1 \).
7. Problem: Combinatorics¶
A standard deck of 52 playing cards is shuffled, and the top 13 cards are dealt out. What is the probability that there are exactly 4 aces among them?
8. Problem: Number Theory¶
Find the smallest positive integer \( n \) such that \( 2^n + 3^n \) is divisible by 5.
9. Problem: Geometry and Circles¶
Two circles of radius 5 are tangent to each other externally. Find the length of the common external tangent.
10. Problem: Algebra¶
Solve for \( x \) if \( x + \frac{1}{x} = 3 \).
11. Problem: Sequences and Series¶
A geometric sequence has the first term \( a_1 = 5 \) and the common ratio \( r = 3 \). Find the sum of the first 6 terms of the sequence.
12. Problem: Combinatorics¶
How many distinct 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 if no digit repeats?
These problems reflect the difficulty level and style typical of the AIME. The solutions usually require multiple steps, creativity, and a deep understanding of various mathematical concepts. Each answer to an AIME problem is a non-negative integer between 0 and 999, which distinguishes it from multiple-choice formats like the AMC.
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