Practice (90/1000)
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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 $$\sum_{n=1}^{\infty}\frac{2}{n^2 + 4n +3}$$
Compute $$\sum_{k=1}^{\infty}\frac{1}{k^2 + k}$$
Compute the value of $$\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}$$
Compute the value of $\sqrt{1+1995\sqrt{4+1995\times 1999}}$.
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Find the length of the leading non-repeating block in the decimal expansion of $\frac{2017}{3\times 5^{2016}}$. For example, the length of the leading non-repeating block of $\frac{1}{6}=0.1\overline{6}$ is 1.
Show that $1\cdot 1! + 2\cdot 2! + \cdots + n\cdot n! = (n+1)!-1$
Compute $$\binom{2022}{1} - \binom{2022}{3} + \binom{2022}{5}-\cdots + \binom{2022}{2021}$$
As shown, points $X$ and $Y$ are on the extension of $BC$ in $\triangle{ABC}$ such that the order of these four points are $X$, $B$, $C$, and $Y$. Meanwhile, they satisfy the relation $BX\cdot AC = CY\cdot AB$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle{ACX}$ and $\triangle{ABY}$, respectively. If $O_1O_2$ intersects $AB$ and $AC$ at $U$ and $V$, respectively, show that $\triangle{AUV}$ is isosceles.
