Let $z$ be a complex number, and $|z|=1$. Find the maximal value of $u=|z^3-3z+2|$.
Let integer $n\ge 2$, prove $$\sin{\frac{\pi}{n}}\cdot\sin{\frac{2\pi}{n}}\cdots\sin{\frac{(n-1)\pi}{n}}=\frac{n}{2^{n-1}}$$
Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.
Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$
Prove: $(x-1) | P(x)$
Let $f(z) = z^2 + az + b$, where both $a$ and $b$ are complex numbers. If for all $|z|=1$, find the values of $a$ and $b$.
If $x$ and $y$ are positive integer solutions to the equation $x^2 - 2y^2 = 1$, then $6\mid xy$.
Let $\alpha$ and $\beta$ be two real roots of the equation $x^2 + x - 4=0$. Find the value of $\alpha^2 - 5\beta + 10$ without computing the value of $\alpha$ and $\beta$.
Let three non-zero numbers $a$, $b$, and $c$ satisfy $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}$. Prove at least two of these three numbers are opposite numbers
If $a_0, a_1,\cdots, a_n \in \{0, 1, 2,\cdots, 9\}, n\ge 1, a_0\ge 1$, then the zeros of $f(x)=a_0 x^n + a_1x^{n-1} +\cdots +a_n$ have real parts less then 4.
Simplify $\displaystyle\frac{2^2-2}{2^2+2}\cdot\displaystyle\frac{3^2-3}{3^2+3}\cdots\displaystyle\frac{10^2-2}{10^2+10}$
Let $a$, $b$ be real numbers such that $ab=-1$, $a+b=3$, compute $a^3+b^3$.
Compute $\Large(\sqrt{6+4\sqrt{2}} + \sqrt{6-4\sqrt{2}}\Large)^2$
Find the roots of $27x^3 + 9x^2 -30+8$
Denote $a$ and $b$ the roots of $(x-2)(x+4)+(x-3)(x+5)-(x-2)(x+5)=0$. Computer $a^3 + b^3 + \frac{11}{(a-1)(b-1)}$
Let $a$, $b$, $c$ be distinct nonzero real numbers, such that $$a+\frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$$
Prove that $|abc|=1$.
The Fibonacci numbers are defined by $F_1=1, F_2=1$, and $F_n=F_{n-1} + F_{n-2}$ for $n=3, 4, \cdots$. Find and prove a formula for the sum of the first $n$ Fibonacci numbers, i.e. $F_1 + F_2 + \cdots +F_n$.
Solve the following equations for all real numbers $r, s, t$:
$$
\begin{array}{rl}
rst &=30\\
rs+st+tr &=-11\\
r+s+t &=-4
\end{array}
$$
Find all real solutions to the equations:
$$
\begin{array}{rl}
(x-y)(x^3+y^3)&=7\\
(x+y)(x^-y^3)&=3
\end{array}
$$
Let $x, y$ and $z$ be real numbers such that
$$
\begin{array}{rl}
x^2 +2(y-1)(z-1) &=12 \\
y^2 +2(z-1)(x-1) &=6 \\
z^2 +2(x-1)(y-1) &=9
\end{array}
$$
Find all the possible values of $x+y+z$.
Find all polynomials $f(x)$ such that $f(x^2) = f(x)f(x+1)$.
Let $\gamma_i$ and $\overline{\gamma_i}$ be the 10 zeros of $x^{10}+(13x-1)^{10}$, where $i=1, 2, 3, 4, 5$. Compute $$\frac{1}{\gamma_1 \overline{\gamma_1}}+\frac{1}{\gamma_2 \overline{\gamma_2}}+\cdots+\frac{1}{\gamma_5 \overline{\gamma_5}}$$
Let {$a_n$} be a sequence with $a_1=1$. If for any $n > 1$, $a_n$ equals one plus twice of the sum of all the previous terms, express $a_n$ in terms of $n$.
Let $x$ and $y$ be real numbers such that $$2 < \frac{x-y}{x+y} < 5$$ If $\frac{x}{y}$ is an integer, what is its value?
What is the area of region bounded by the graphs of $y=|x+2| -|x-2|$ and $y=|x+1|-|x-3|$?
Let $S$ be the sum of all distinct real solutions of the equation $$\sqrt{x+2015}=x^2-2015$$
Compute $\lfloor 1/S \rfloor$.