Factorize: $f(x,y,z)=(x+y+z)^5-x^5-y^5-z^5$
Factorize $f(x,y,z) = x^3+y^3 +z^3 - 3xyz$.
Simplify $$\frac{(y-z)^3 +(z-x)^3+(x-y)^3}{(y-z)(z-x)(x-y)}$$
If equation $x^2 - (1-2a)x+a^2-3 = 0$ has two distinct real roots, and equation $x^2 -2x+2a-1=0$ is not solvable in real numbers, find the values of $a$ such that the roots of the first equation are integers.
If one root of the equation $x^2 -6x+m^2-2m+5=0$ is $2$. Find the value of the other root and $m$.
If the equation $x^2+2(m-2)x + m^2 + 4 = 0 $ has two real roots, and the sum of their square is 21 more than their product, find the value of $m$.
Let $\alpha$ and $\beta$ be the two roots of $x^2 + 2x -5=0$. Evaluate $\alpha^2 + \alpha\beta + 2\alpha$.
If at least one real root of equation $x^2 - mx +5+m=0$ equals one root of $x^2 - (7m+1)x+13m+7=0$, compute the product of the four roots of these two equations.
If the difference of the two roots of the equation $x^2 + 6x + k=0$ is 2, what is the value of $k$?
If the two roots of $(a^2 -1)x^2 -(a+1)x+1=0$ are reciprocal, find the value of $a$.
Let $x_1$ and $x_2$ be the two roots of $x^2 - 3mx +2(m-1)=0$. If $\frac{1}{x_1}+\frac{1}{x_2}=\frac{3}{4}$, what is the value of $m$?
Let $x_1$ and $x_2$ be the two roots of $2x^2 -7x -4=0$, compute the values of the following expressions using as many different ways as possible.
(1) $x_1^2 + x_2^2$
(2) $(x_1+1)(x_2+1)$
(3) $\mid x_1 - x_2 \mid$
If one root of $x^2 + \sqrt{2}x + a = 0$ is $1-\sqrt{2}$, find the other root as well as the value of $a$.
Consider the equation $x^2 +(m-2)x + \frac{1}{2}m-3=0$.
(1) Show that this equation always have two distinct real roots
(2) Let $x_1$ and $x_2$ be its roots. If $x_1+x_2=m+1$, what is the value of $m$?
If $n>0$ and $x^2 -(m-2n)x + \frac{1}{4}mn=0$ has two equal positive real roots, what is the value of $\frac{m}{n}$?
If real number $m$ and $n$ satisfy $mn\ne 1$ and $19m^2+99m+1=0$ and $19+99n+n^2=0$, what is the value of $\frac{mn+4m+1}{n}$?
Let $x_1$ and $x_2$ be two real roots of $m^2x^2 +2(3-m)x+1=0$. If $m=\frac{1}{x_1}+\frac{1}{x_2}$, what is the value of $m$?
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2(k+1)x+k^2 + 2 = 0$. If $(x_1+1)(x_2+1) =8$, find the value of $k$
Let $x_1$ and $x_2$ be the two real roots of the equation $x^2 - 2mx + (m^2+2m+3)=0$. Find the minimal value of $x_1^2 + x_2^2$.
How many integer pairs $(a,b)$ with $1 < a, b\le 2015$ are there such that $log_a b$ is an integer?
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n =\frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
A word is an ordered, non-empty sequence of letters, such as word or wrod. How many distinct words can be made from a subset of the letters $\textit{c, o, m, b, o}$, where each letter in the list is used no more than the number of times it appears?
What is the $22^{nd}$ positive integer $n$ such that $22^n$ ends in a $2$?
Find the sum of all positive integers $n$ such that the least common multiple of $2n$ and $n^2$ equals $(14n - 24)$?
What is the largest positive integer $n$ less than $10,000$ such that in base 4, $n$ and $3n$ have the same number of digits; in base 8, $n$ and $7n$ have the same number of digits; and in base 16, $n$ and $15n$ have the same number of digits? Express your answer in base 10.