Algebraic Manipulations: all problems

Assuming $a \neq 3$, $b \neq 4$, and $c \neq 5$, what is the value in simplest form of the following expression?

Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y} = -2$. What is the value of $\frac{x+3y}{3x-y}$?

Real numbers $a$ and $b$ satisfy the equations $3^a = 81^{b+2}$ and $125^b = 5^{a-3}$. What is $ab$?

Let $a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5$. What is $a + b + c + d$?

If $y + 4 = (x - 2)^2$, $x + 4 = (y - 2)^2$, and $x \neq y$, what is the value of $x^2 + y^2$?

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of

For what value $x$ does $10^{x} \cdot 100^{2x} = 1000^{5}$?

Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{2}{x} = y + \frac{2}{y}$, what is $xy$?

Suppose that real number $x$ satisfies

What is the value of $\sqrt{49 - x^2} + \sqrt{25 - x^2}$?

Two non-zero real numbers, $a$ and $b$, satisfy $ab = a - b$. Which of the following is a possible value of $\frac{a}{b} + \frac{b}{a} - ab$?

Real numbers $x$ and $y$ satisfy the equation $x^2 + y^2 = 10x - 6y - 34$. What is $x + y$?

If $x + \frac{1}{x} = 4$, then what is the value of $x^8 + \frac{1}{x^8}$?

If $\left(a - \frac{1}{a}\right)^2 = 4$, what is the absolute value of $\left(a^3 - \frac{1}{a^3}\right)$?

Which of the following is equivalent to $\sqrt{\dfrac{x}{1 - \frac{x-1}{x}}}$ when $x < 0$?

Find $x^2 + y^2$ if $x$ and $y$ are positive integers such that

Let $x_1 \leq x_2 \leq \cdots \leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$. Among all such 100-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Suppose that $x$, $y$, and $z$ are three positive numbers that satisfy the equations $xyz = 1$, $x + \frac{1}{z} = 5$, and $y + \frac{1}{x} = 29$. Then $z + \frac{1}{y} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Find $ax^5 + by^5$ if the real numbers $a$, $b$, $x$, and $y$ satisfy the equations

The equation $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.

What is the product of the real roots of the equation $x^2 + 18x + 30 = 2\sqrt{x^2 + 18x + 45}$?

Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation:

Let $a$, $b$, $c$, and $d$ be real numbers that satisfy the system of equations

There exist relatively prime positive integers $m$ and $n$ such that

Find $m + n$.

Let $m$ be the largest real solution to the equation

There are positive integers $a$, $b$, and $c$ such that $m = a + \sqrt{b + \sqrt{c}}$. Find $a + b + c$.

Let $x$ and $y$ be real numbers satisfying $x^4y^5 + y^4x^5 = 810$ and $x^3y^6 + y^3x^6 = 945$. Evaluate $2x^3 + (xy)^3 + 2y^3$.

The equation $2000x^6 + 100x^5 + 10x^3 + x - 2 = 0$ has exactly two real roots, one of which is $\frac{m + \sqrt{n}}{r}$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r > 0$. Find $m + n + r$.