Practice (TheColoringMethod)
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If $x^2 +x =m = (x-n)^2$, what is the value of $4(m+n)$?
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 $\frac{a}{b} =\frac{3}{5}$ and $b$ = 10, what is the value of $a$?
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 $3x^2+12x-5$.
Square ABCD, shown here, has diagonals AC and BD that intersect at E. How many triangles of any size are in the figure?
If $x+y=4$ and $x^2+y^2=6$, find the value of $xy$.
When the integers $1$ to $100$ inclusive are written, what digit is written the fewest number of times?
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 $3x^6-3x^2$.
Factorize $x^2-4xy-1+4y^2$.
Factorize $ax^2 -bx^2 -bx + ax +b-a$.
Factorize $9x^4-36y^2$.
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 $(a^2-b^2)-8(a^2+b^2)$
Danica wrote the digits from 1 to 8 across a sheet of paper, as shown, and then circled one digit. If the digits to the left of the circled digit had the same sum as those to the right of the circled digit, which digit did Danica circle?
Sally received a sum of money for her birthday. She spent one-third of that money on bus fare to travel downtown. She spent half of the money that remained to treat her best friend to a movie, after which Sally had $12.00 remaining. How much money did Sally receive for her birthday?
When six is added to a number, the result is three times the original number. What is the original number?
Each time one of Yuan's paintings sells, the gallery sends Yuan a payment equal to 60% of the purchase amount, and the gallery keeps the remainder. If three of Yuan's paintings sold recently for \$200, \$300 and \$500, he should expect to receive three payments totaling how many dollars from his gallery?
If $x$ is 15% of 500 and $y$ is 200% of $x$, what is the value of the sum $x + y$\u200a?
The square in Figure 1 is cut along its diagonals creating four congruent triangles that then are arranged to create Figure 2. What is the probability that a randomly chosen point within the boundary of Figure 2 is in any of the shaded triangles? Express your answer as a common fraction.
If $\sqrt[3]{b} = 5$, what is the value of $2b$?