Solve this equation $$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$$
Let non-zero real numbers $a, b, c$ satisfy $a+b+c\ne 0$. If the following relations hold $$\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$$
Find the value of $$\frac{(a+b)(b+c)(c+a)}{abc}$$
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 $2310$.
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 $a, b, c, p$ 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 $abc+p=0$.
Show that if a polynomial $P(x)$ satisfies $P(2x^2-1)=(P(x))^2/2$, then it must be a constant.
Let $d\ne 0$ be the common difference of an arithmetic sequence $\{a_n\}$, and positive rational number $q < 1$ be the common ratio of a geometric sequence $\{b_n\}$. If $a_1=d$, $b_1=d^2$, and $\frac{a_1^2+a_2^2+a_3^2}{b_1+b_2+b_3}$ is a positive integer, what is the value of $q$?
Let real numbers $a, b, c, d$ satisfy
$$
\left\{
\begin{array}{ccl}
ax+by&=3\\
ax^2+by^2&=7\\
ax^3+by^3&=16\\
ax^4 + by^4 &=42
\end{array}
\right.
$$
Find $ax^5+by^5$.
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 $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
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: $$\frac{2a^2}{1+a^2}=b,\qquad \frac{2b^2}{1+b^2}=c,\qquad \frac{2c^2}{1+c^2}=a$$
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 $f(1)$?
Let $f(x)=x^3 -x^2 -13x+24$. Find three pairs of $(x,y)$ such that if $y=f(x)$, then $x=f(y)$.