For $n\ge 1$, let $d_n$ denote the length of the line segment connecting the two points where the line $y = x + n + 1$ intersects the parabola $8x^2 = y - \frac{1}{32}$ . Compute the sum $$\sum_{n=1}^{1000}\frac{1}{n\cdot d_n^2}$$
Determine all pairs $(a, b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.
Let $$f(r) = \displaystyle\sum_{j=2}^{2008}\frac{1}{j^r} = \frac{1}{2^r}+\frac{1}{3^r}+\cdots+\frac{1}{2016^r}$$
Find $$\sum_{k=2}^{\infty}f(k)$$
Let $P(x)$ be a polynomial with degree 2008 and leading coeffi\u000ecient 1 such that $P(0) = 2007, P(1) = 2006, P(2) = 2005, \cdots, P(2007) = 0$. Determine the value of $P(2008)$. You may use factorials in your answer.
Evaluate the infinite sum $\displaystyle\sum_{n=1}^{\infty}\frac{n}{n^4+4}$.
Solve the equation $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}-\sqrt{x}=1$$
Find all $x$ such that $\displaystyle\sum_{k=1}^{\infty}kx^k=20$.
Find the length of the leading non-repeating block in the decimal expansion of $\frac{2004}{7\times 5^{2003}}$. For example the length of the leading non-repeating block of $\frac{5}{12}=0.41\overline{6}$ is $2$.
Find $x$ satisfying $x=1+\frac{1}{x+\frac{1}{x+\cdots}}$.
Write $\sqrt[3]{2+5\sqrt{3+2\sqrt{2}}}$ in the form of $a+b\sqrt{2}$ where $a$ and $b$ are integers.
Simplify $$\sin{x} + \sin{2x} + \cdots +\sin{nx}$$ and $$\cos{x} + \cos{2x} + \cdots + \cos{nx}$$
Solve the equation $\cos\theta + \cos 2\theta + \cos 3\theta = \sin \theta +\sin 2\theta + \sin 3\theta$.
Let $A, B,$ and $C$ be angles of a triangle. If $\cos 3A + \cos 3B + \cos 3C = 1$, determine the largest angle of the triangle.
Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{a_k^2+b_k^2}$ where $\{a_k\}$ is an arithmetic sequence whose first term is $4$ and common difference is $8$. $b_1, b_2,\cdots, b_n$ are positive real numbers satisfying $\displaystyle\sum_{k=1}^nb_k=17$. If there exist a positive integer $n$ such that $S_n$ is also an integer, find $n$.
Find the value of $\cos 20^\circ \cos 40^\circ \cos 80^\circ$ using at least two difference methods.
Simplofy $\sin\theta + \frac{1}{2}\cdot\sin 2\theta + \frac{1}{4}\cdot\sin 3\theta + \cdots$.
Solve the equation $\cos^2 x + \cos^2 2x +\cos^2 3x=1$ in $(0, 2\pi)$.
Prove for every positive integer $n$ and real number $x\ne \frac{k\pi}{2^t}$ where $t =0, 1, 2,\cdots$ and $k$ is an integer, the following relation always holds:
$$\frac{1}{\sin 2x}+\frac{1}{\sin 4x} + \cdots +\frac{1}{\sin 2^nx}=\frac{1}{\tan x}-\frac{1}{\tan 2^nx}$$
Let $\alpha\in\Big(\frac{3\pi}{2}, 2\pi\Big)$. Simplify $$\sqrt{\frac{1}{2}-\frac{1}{2}\sqrt{\frac{1}{2}+\frac{1}{2}\cdot\cos 2\alpha}}$$
Without using a calculator, find the value of $\cos\frac{\pi}{13}+\cos\frac{3\pi}{13}+\cos\frac{9\pi}{13}$.
Simplify $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} +\binom{n}{2} - \frac{1}{2}\binom{n}{3} + \cdots $$
Let $n > k$ be two positive integers. Simplify the following expression $$\binom{n}{k} + 2\binom{n-1}{k} + 3\binom{n-2}{k} + \cdots+ (n-k+1)\binom{k}{k}$$
Let positive integers $m$ and $n$ satisfy $m\le n$. Prove $$\sum_{k=m}^n\binom{n}{k}\binom{k}{m}=2^{n-m}\binom{n}{m}$$
Show that $$\sum_{k=0}^{2n-1}(-1)^k(k+1)\binom{2n}{k}^{-1}=\frac{1}{\binom{2n}{0}}-\frac{2}{\binom{2n}{1}}+\cdots-\frac{2n}{\binom{2n}{2n-1}}=0$$
Compute $$\sum_{n=1}^{\infty}\frac{2}{n^2 + 4n +3}$$