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.
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.
Let $a$ and $b$ be the two roots of $x^2 - 3x -1=0$. Try to solve the following problems without computing $a$ and $b$:
1) Find a quadratic equation whose roots are $a^2$ and $b^2$
2) Find the value of $\frac{1}{a+1}+\frac{1}{b+1}$
3) Find the recursion relationship of $x_n=a^n + b^n$
Find as many different solutions as possible.
Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.
In $\triangle{ABC}$, let $a$, $b$, and $c$ be the lengths of sides opposite to $\angle{A}$, $\angle{B}$ and $\angle{C}$, respectively. $D$ is a point on side $AB$ satisfying $BC=DC$. If $AD=d$, show that
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
Suppose $a_1$, $b_1$, $c_1$, $a_2$, $b_2$, and $c_2$ are all positive real numbers. If both $a_1x^2 +b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=$ are solvable in real numbers. Show that their roots must be all negative. Furthermore, prove equation $a_1a_2x^2+b_1b_2x+c_1c_2=0$ has two negative real roots too.
Let $x$, $y$, and $z$ be real numbers satisfying $x=6-y$ and $z^2=xy-9$. Show that $x=y$.
Let $\alpha_n$ and $\beta_n$ be two roots of equation $x^2+(2n+1)x+n^2=0$ where $n$ is a positive integer. Evaluate the following expression $$\frac{1}{(\alpha_3+1)(\beta_3+1)}+\frac{1}{(\alpha_4+1)(\beta_4+1)}+\cdots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}$$
Let real numbers $a$, $b$, and $c$ satisfy
$$
\left\{
\begin{array}{rcl}
a^2 - bc-8a +7&=&0\\
b^2 + c^2 +bc-6a+6&=&0
\end{array}
\right.
$$
Show that $1 \le a \le 9$.
Find one real solution $(a, b, c, d)$ to the following system:
$$
\left\{
\begin{array}{rcl}
a+b+c+d&=&-2\\
ab+ac+ad+bc+bd+cd&=&-3\\
abc+abd+acd+bcd&=&4\\
abcd&=&3
\end{array}
\right.
$$
If $m^2 = m+1, n^2-n=1$ and $m\ne n$, compute $m^7 +n^7$.
Find the range of real number $a$ if the two roots of $x^2+2ax+6-a=0$ satisfy one of the following condition:
- two roots are both greater than 1
- one root is greater than 1 and the other is less than 1
Solve equation $(6x+7)^2(3x+4)(x+1)=6$ in real numbers.
If $x^2 + 11x+16=0, y^2 + 11y+16=0$, and $x\ne y$, what is the value of $$\sqrt{\frac{x}{y}}-\sqrt{\frac{y}{x}}$$
Let $x_1$ and $x_2$ be two real roots of $x^2-x-1=0$. Find the value of $2x_1^5 + 5x_2^3$.
Find integer $m$ such that the equation $x^2+mx-m+1=0$ has two positive integer roots.