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.