If $x^m - y^n = (x+y^2)(x-y^2)(x^2+y^4)$, find the value of $m+n$.
If $1+x+x^2+\cdots + x^{2014}+x^{2015}=0$, find the value of $x^{2016}$.
If the value of $(9x^2 + k + y^2)$ is a perfect square for any $x$ and $y$, what value can $k$ take?
If $x^2+4x-4=0$, find the value of $3x^2+12x-5$.
If $x+y=4$ and $x^2+y^2=6$, find the value of $xy$.
Evaluate the value of $$\Big(1-\frac{1}{2^2}\Big)\Big(1-\frac{1}{3^2}\Big)\cdots\Big(1-\frac{1}{9^2}\Big)\Big(1-\frac{1}{10^2}\Big)$$
Factorize: $x^4-2x^3-35x^2$
Factorize $x^2-4xy-1+4y^2$.
Factorize $ax^2 -bx^2 -bx + ax +b-a$.
Factorize $(x+1)(x+2)(x+3)(x+4)-24$.
Prove: for any given positive integer $n$, the value of $(n+7)^2 -(n-5)^2$ must be a multiple of 24.
If $a+b=2$, find the value of $(a^2-b^2)-8(a^2+b^2)$
If A+B = 5 and A-B = 3, what is the value of A+A\u200a?
Working alone, a professor grades a paper every 10 minutes. The professor spends 20 minutes training an assistant. Then, working together, they grade 2 papers every 15 minutes. For how many graded papers is the amount of time it would take the professor working alone the same as the amount of time it would take the professor and her assistant working together, including the time required for training?
If $b = a^2$ and $c = 3b - 2$, what is the product of all values of $a$ for which $b = c$?
If $f$ is a function such that $f(f(x)) = x^2 - 1$, what is $f(f(f(f(3))))$?
If the sum of an arithmetic progression of six positive integer terms is 78, what is the greatest possible difference between consecutive terms?
For a particular sequence $a_1 = 3$, $a_2 = 5$ and $a_n = a_{n -1} - a_{n -2}$, for $n \ge 3$. What is the $2015^{th}$ term in this sequence?
When $\frac{1}{98}$ is expressed as a decimal, what is the $10^{th}$ digit to the right of the decimal point?
What is the correct ordering of the three numbers, $10^8$, $5^{12}$, and $2^{24}$?
What time was it $2011$ minutes after midnight on January $1$, $2011$?
The taxi fare in Gotham City is \$2.40 for the first $\frac12$ mile and additional mileage charged at the rate \$0.20 for each additional 0.1 mile. You plan to give the driver a \$2 tip. How many miles can you ride for \$10?
Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \cdot 3^x \cdot 5^y \cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?