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You have shared a mathematical problem sheet that contains multiple exercises requiring proofs. I can assist with these proofs or guide you through solving any specific problem. Here is an overview of the tasks:

1. Prove that:
   \[
   \sin x + \sin 2x + \dots + \sin nx = \sin \frac{nx}{2} \frac{\sin \frac{n+1}{2}x}{\sin \frac{x}{2}}
   \]
   
2. Prove the inequality:
   \[
   2^n \geq 2n + 1
   \]
   
3. Prove that:
   \[
   \frac{4^n}{n+1} \leq \frac{(2n)!}{(n!)^2}
   \]
   
4. Prove that:
   \[
   (1 + \alpha)(1 + \beta) \cdots (1 + \lambda) > 1 + \alpha + \beta + \dots + \lambda
   \]
   where all \(\alpha, \beta, \dots > -1\).

5. Prove Chebyshev's inequality:
   \[
   \frac{x_1 y_n + x_2 y_{n-1} + \dots + x_n y_1}{n} \leq \frac{x_1 y_1 + x_2 y_2 + \dots + x_n y_n}{n}
   \]
   given that \(x_1 \geq x_2 \geq \dots \geq x_n\) and \(y_1 \geq y_2 \geq \dots \geq y_n\).

6. Given \(x > 0\), prove the inequality:
   \[
   \frac{x^n}{1 + x + x^2 + \dots + x^{2n}} \leq \frac{1}{2n+1}
   \]

7. Given the positive numbers \(a, b, c, d\) with product 1, prove the inequality:
   \[
   \frac{1}{ab+bc+cd-1} + \frac{1}{bc+cd+da-1} + \frac{1}{cd+da+ab-1} + \frac{1}{da+ab+bc-1} \leq 2
   \]

Let me know which problem you would like help with, or if you prefer a detailed explanation for any of these! 

Here are five related questions that could help you deepen your understanding:

1. How can summation formulas simplify trigonometric series?
2. What is the relation between binomial coefficients and combinatorial inequalities like the one in problem 3?
3. How does the assumption that \(\alpha, \beta, \dots > -1\) influence inequality proofs?
4. What is the significance of ordering sequences \(x_1 \geq x_2 \geq \dots \geq x_n\) in Chebyshev-type inequalities?
5. How do methods from number theory help in proving inequalities for positive numbers?

**Tip:** When dealing with inequalities, try exploring symmetry or using mathematical induction for efficient proof techniques.

Prove the following mathematical expressions and inequalities involving trigonometric series, binomial coefficients, inequalities, and Chebyshev's inequality.

This problem set involves advanced mathematical proofs including trigonometric series, combinatorial inequalities, and Chebyshev's inequality. Suitable for university-level students, these exercises cover key concepts in trigonometry, inequalities, and combinatorics.

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