The equations $2x + 7 = 3$ and $bx - 10 = - 2$ have the same solution. What is the value of $b$?
There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$. What is the sum of those values of $a$?
The quadratic equation $x^2+mx+n=0$ has roots twice those of $x^2+px+m=0$, and none of $m,n,$ and $p$ is zero. What is the value of $\frac{n}{p}$?
Let the sequence {$a_n$} satisfy $a_0=0$, $a_1=1$, $a_{n+2} = (n+3)a_{n+1} -(n+2)a_n$. Find whether the following equation is solvable in rational numbers:$$\sum_{i=1}^n\frac{x^i}{a_i-a_{i-1}}=-1\qquad\qquad(n \ge 2)$$
If the sum of two numbers is 4 and their difference is 2, what is their product?
What is the value of $\frac{444^2-111^2}{444-111}$ ?
Real numbers a and b satisfy the equation $\frac{2a-4}{5}+\frac{3a+1}{5}=b$. What is the value of $a - b$? Express your answer as a common fraction.
If $4(a - 3) - 2(b + 5) = 14$ and $5b -a = 0$, what is the value of $a + b$?
The product of two consecutive integers is five more than their sum. What is the smallest possible sum of two such consecutive integers?
If $40q = p + \frac{p}{3}+\frac{p}{9}+\frac{p}{27}$ , what is the ratio $\frac{q}{p}$? Express your answer as a common fraction.
What is the sum of all real numbers $x$ such that $4^x - 6 \times 2^x + 8 = 0$?
Seven pounds of Mystery Meat and four pounds of Tastes Like Chicken cost \$78.00. Tastes Like Chicken costs \$3.00 more per pound than Mystery Meat. In dollars, how much does a pound of Mystery Meat cost?
Jay and Mike were walking home with heavy books in their backpacks. When Mike complained about the weight in his backpack, Jay remarked, "If I take one of your books, I will be carrying twice as many books as you will be carrying, but if you take one of my books, we'll be carrying the same number of books." How many books is Mike carrying in his backpack?
The sum of the squares of two positive numbers is 20 and the sum of their reciprocals is 2. We must find their product.
Adult tickets are \$5 and student tickets are \$2. Five times as many student tickets were sold as adult tickets for a total of \$1125. We must find the number of tickets sold.
The positive difference of the cubes of two consecutive positive integers is 111 less than 5 times the product of the two consecutive integers. We must find the sum of the two consecutive integers.
If $x^2+\frac{1}{x^2}= 3$, then what is the value of $\frac{x^2}{(x^2+1)^2}$?
For how many ordered pairs $(x, y)$ of integers satisfying $0 \le x$, $y \le 10$, and $(x + y)^2 + (xy - 1)^2$ is a prime number?
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 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 $b$.
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.