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ThePerfectHacker · #1 March 1st, 2008, 06:13 PM

1a)Let $n$ be an odd positive integer. Let $\zeta = e^{2\pi i/n}$ prove that $x^n - y^n = \prod_{k=0}^{(n-1)} (x\zeta^k - y\zeta^{-k})$ .
1b)Let $f(z) = 2i\sin (2\pi z)$ and $n$ an odd positive integer. Prove that $f(nz) = f(z)\prod_{k=1}^{(n-1)/2} f\left( z + \frac{k}{n} \right) f\left(z - \frac{k}{n} \right)$ .

2)Each man of $n\geq 2$ 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 $\text{C}_1$ and $\text{C}_2$ be circles with $\text{C}_1$ inside $\text{C}_2$ and tangent at a point on $\text{C}_2$ . Let $c_1,c_2,...,c_k$ ( $k\geq 3$ ) be circles in between $\text{C}_1$ and $\text{C}_2$ and tangent to $\text{C}_1$ and $\text{C}_2$ and to eachother adjacent circle. Let $a_1,a_2,...,a_{k-1}$ be the points of tangency of these circles with their neighbors. Prove that $a_1,a_2,...,a_{k-1}$ all lie on a common circle.
(Note: The solution I know does not use elementary geometry).