https://r2.erweima.ai/mathbot/image/17259922185940lrm41nn.pdf

You have provided a PDF from the *International Youth Math Challenge* with several interesting math problems from the 2024 qualification round. Here is a breakdown of the problems:

### Problem A
**Continue the two sequences of numbers and find an equation for the n-th term:**
- Sequence \( a_n: 2, 5, 10, 17, 26, 37, \ldots \)
- Sequence \( b_n: 1, 2, 8, 48, 384, 3840, \ldots \)

### Problem B
**Solve the equation for all real \( x \):**
\[
x^4 + x^2 - x - 1 = 1 - x - x^2 - x^4
\]

### Problem C
**Evaluate the following expression without using a calculator:**
\[
\left( \sqrt{2} + \left( 32 \right)^{1/4} + \sum_{m=1}^{3} \left( \frac{1}{m!} - \sqrt{m} \right) \right) \cdot \left( 2 \log_2(8) + \frac{1}{23} - \prod_{k=1}^{8} \left( 1 + \frac{1}{k} \right) \right)
\]

### Problem D
**Prove that \( n^5 - n \) is divisible by 5 for all positive integers \( n \).**

### Problem E
**Find the surface area of the rotated square that is not overlapping the non-rotated square.**

Would you like to explore a specific problem or need a detailed solution for one of them?

Here are five related questions:
1. How do you derive a general formula for sequences like in Problem A?
2. What strategies can help solve quartic equations like in Problem B?
3. What simplification techniques are useful for expressions involving roots and factorials, as seen in Problem C?
4. What methods can be used to prove divisibility statements like in Problem D?
5. How can geometric transformations affect area, as in Problem E?

**Tip:** When working on complex sequences, try finding the differences between terms to identify patterns or functions!

You have provided a PDF from the *International Youth Math Challenge* with several interesting math problems from the 2024 qualification round. The problems include sequences, equations, expressions, divisibility, and geometry.

Explore challenging math problems from the 2024 International Youth Math Challenge qualification round, including sequence derivation, solving quartic equations, and proving divisibility theorems. This set of problems is ideal for students in Grades 9-12 preparing for advanced mathematical competitions.

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