Let real numbers $a$, $b$, and $c$ satisfy $a+b+c=2$ and $abc=4$. Find
the minimal value of the largest among $a$, $b$, and $c$.
the minimal value of $\mid a\mid +\mid b \mid +\mid c \mid$.
If $a\ne 0$ and $\frac{1}{4}(b-c)^2=(a-b)(c-a)$, compute $\frac{b+c}{a}$.
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.
Let real numbers $a, b, c$ satisfy $a > 0$, $b>0$, $2c>a+b$, and $c^2>ab$. Prove $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$
Suppose the graph of $f(x)=x^4 + ax^3 + bx^2 + cd + d$, where $a$, $b$, $c$, $d$ are all real constants, passes through three points $A \big(2,\frac{1}{2}\big)$, $B \big(3, \frac{1}{3}\big)$, and $C \big(4, \frac{1}{4}\big)$. Find the value of $f(1) + f(5)$.
Find a quadratic polynomial $f(x)=x^2 + mx +n$ such that $$f(a)=bc,\quad f(b) = ca,\quad f(c) = ab$$ where $a$, $b$, $c$ are three distinct real numbers.
If all coefficients of the polynomial $$f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_3x^3+x^2+x+1=0$$ are real numbers, prove that its roots cannot be all real.
Compute the value of $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}+\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
and simplify $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
Prove there cannot exist a $998$-degree polynomial with real number coefficients $P(x)$ such that $$[P(x)]^2-1=P(x^2+1)$$ holds for any $x\in\mathbb{C}$.
If $abc=1$, solve this equation $$\frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1}=1$$
How many pairs of ordered real numbers $(x, y)$ are there such that
$$
\left\{
\begin{array}{ccl}
\mid x\mid + y &=& 12\\
x + \mid y \mid &=&6
\end{array}
\right.
$$
Let $a$, $b$, and $c$ be three distinct numbers such that $$\frac{a+b}{a-b}=\frac{b+c}{2(b-c)}=\frac{c+a}{3(c-a)}$$
Prove that $8a + 9b + 5c = 0$.
Solve this equation in real numbers: $$\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{1}{2}\times(x+y+z)$$
Let $a$, $b$, and $c$ be the lengths of $\triangle{ABC}$'s three sides. Compute the area of $\triangle{ABC}$ if the following relations hold: $$\frac{2a^2}{1+a^2}=b,\qquad \frac{2b^2}{1+b^2}=c,\qquad \frac{2c^2}{1+c^2}=a$$
Let real numbers $x$, $y$, and $z$ satisfy $0 < x, y, z < 1$. Prove $$x(1-y)+y(1-z)+z(1-x)< 1$$
Show that $$\frac{(x+a)(x+b)}{(c-a)(c-b)}+\frac{(x+b)(x+c)}{(a-b)(a-c)}+\frac{(x+c)(x+a)}{(b-c)(b-a)}=1$$ without expanding the left side of the equation.
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.
If sequence $\{a_n\}$ has no zero term and satisfies that, for any $n\in\mathbb{N}$, $$(a_1+a_2+\cdots+a_n)^2=a_1^3+a_2^3+\cdots+a_n^3$$
- Find all qualifying sequences $\{a_1, a_2, a_3\}$ when $n=3$.
- Is there an infinite sequence $\{a_n\}$ such that $a_{2013}=-2012$? If yes, give its general formula of $a_n$. If not, explain.
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\]has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\]What is $f(1)$?
We define the Fibonaccie numbers by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n}$. Find the greatest common divisor $(F_{100}, F_{99})$, and $(F_{100}, F_{96})$.
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.
The function $f$ satisfies $f(0)=0$, $f(1)=1$, and $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ for all $x,y\in\mathbb{R}$. Show that $f(x)=x$ for all rational numbers $x$.
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Let $\{a_n\}$ be a sequence defined as $a_1=1$ and $a_n=\frac{a_{n-1}}{1+a_{n-1}}$ when $n\ge 2$. Find the general formula of $a_n$.