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
$$c+d=2\cdot b\cdot\cos{A}\quad\text{and}\quad c\cdot d = b^2-a^2$$
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
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.
Let $\alpha$ and $\beta$ be two real roots of $x^4 +k=3x^2$ and also satisfy $\alpha + \beta = 2$. Find the value of $k$.
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What probability that a reader guesses at random will match three
- all correctly?
- all incorrectly?
Determine all roots, real or complex, of the following system
\begin{align}
x+y+z &= 3\\
x^2+y^2+z^2 &= 3\\
x^3+y^3+z^3 &= 3
\end{align}
Find all pairs of real numbers $(a, b)$ so that there exists a polynomial $P(x)$ with real coefficients and $P(P(x))=x^4-8x^3+ax^2+bx+40$.
Solve this equation $(x-2)(x+1)(x+4)(x+7)=19$.
Let real numbers $x, y,$ and $z$ satisfy $$x+\frac{1}{y}=4\quad\text{,}\quad y+\frac{1}{z}=1\quad\text{,}\quad z +\frac{1}{x}=\frac{7}{3}$$ Find the value of $xyz$.
Find the range of real number $a$ if equation $\mid\frac{x^2}{x-1}\mid=a$ has exactly two distinct real roots.
Solve this equation $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$
Solve this equation $2x^4 + 3x^3 -16x^2+3x + 2 =0$.
Solve this equation: $(x^2-x-1)^{x+2}=1$.
The sum of two positive integers is $2310$. Show that their product is not divisible by $2310$.
Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.
Suppose $a, b, c$ are all real numbers. If the quadratic polynomial $P(x)=ax^2 + bx + c$ satisfies the condition that $\mid P(x)\mid \le 1$ for all $-1 \le x \le 1$, find the maximum possible value of $b$.
Let $a_1=a_2=1$ and $a_{n}=(a_{n-1}^2+2)/a_{n-2}$ for $n=3, 4, \cdots$. Show that $a_n$ is an integer for $n=3, 4, \cdots$.
Let $a, b, c, p$ be real numbers, with $a, b, c$ not all equal, such that $$a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p$$ Determine all possible values of $p$ and prove $abc+p=0$.