## Instruction
Inspired by the math content, create a math problem appropriate for advanced high school mathematics competitions like the AIME. The problem should challenge students in core areas tested on the AIME such as algebra, geometry, combinatorics, number theory and probability. Encourage creative problem-solving and deep mathematical thinking using pre-calculus level techniques. Avoid delving into college-level topics like abstract/linear algebra, topology, multivariable calculus, or advanced physics. Any physics concepts used should be kept at a basic mechanics level.

## Math Content
<EXTRACT>

## Guidelines 
[Problem]: Design an insightful and challenging problem suitable for AIME participants inspired by the math content. Focus on core AIME topics like algebra, geometry, combinatorics, number theory and probability. The problem should be solvable using clever applications of high school math and mathematical reasoning, without requiring knowledge of advanced college-level mathematics or physics. Aim for a difficulty level on par with actual AIME questions.
[Solution]: Provide a detailed, step-by-step solution that would be enlightening to an advanced high school student, around 250-350 words long. Explain the reasoning process and techniques applied in each step. Use language that is mathematically sophisticated yet understandable to a well-prepared AIME participant. Justify any key insights or creative leaps in the solution process.

## New Problem