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 $b = c$?
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?
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds?
A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box?
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number?
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
The zeroes of the function $f(x)=x^2-ax+2a$ are integers .What is the sum of the possible values of a?
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?
What is the value of $\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}} ?$