1a)Let $\displaystyle n$ be an odd positive integer. Let $\displaystyle \zeta = e^{2\pi i/n}$ prove that $\displaystyle x^n - y^n = \prod_{k=0}^{(n-1)} (x\zeta^k - y\zeta^{-k})$.

1b)Let $\displaystyle f(z) = 2i\sin (2\pi z)$ and $\displaystyle n$ an odd positive integer. Prove that $\displaystyle f(nz) = f(z)\prod_{k=1}^{(n-1)/2} f\left( z + \frac{k}{n} \right) f\left(z - \frac{k}{n} \right)$.

2)Each man of $\displaystyle n\geq 2$ men throws his wallet on the table, then every one picks up a wallet randomly. Find the probability the every person takes the wrong wallet.

3)Let $\displaystyle \text{C}_1$ and $\displaystyle \text{C}_2$ be circles with $\displaystyle \text{C}_1$ inside $\displaystyle \text{C}_2$ and tangent at a point on $\displaystyle \text{C}_2$. Let $\displaystyle c_1,c_2,...,c_k$ ($\displaystyle k\geq 3$) be circles in between $\displaystyle \text{C}_1$ and $\displaystyle \text{C}_2$ and tangent to $\displaystyle \text{C}_1$ and $\displaystyle \text{C}_2$ and to eachother adjacent circle. Let $\displaystyle a_1,a_2,...,a_{k-1}$ be the points of tangency of these circles with their neighbors. Prove that $\displaystyle a_1,a_2,...,a_{k-1}$ all lie on a common circle.

(Note: The solution I knowdoes notuse elementary geometry).