The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?
Show that when $x$ is an integer, $x^2 + 5x + 16$ is not divisible by $169$.
Let $f(x) = x^3+ax^2+bx+c$ have solutions that are distinct negative integers. If $a+b+c =2014$, \ffind $c$.
Evaluate $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\frac{1}{\sqrt{1368}+\sqrt{1369}}$.
$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n)) + f(n) = 2n + 3$ for all nonnegative integers $n$. Find $f(2014)$.
Given that $a_n a_{n-2} - a_{n-1}^2 +a_n-na_{n-2}=-n^2+3n-1$ and $a_0=1$, $a_1=3$, find $a_{20}$.
Solve the equation $$\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=(a+1)\sqrt{\frac{x}{x+\sqrt{x}}}$$
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days?
Which of the following integers cannot be written as the sum of four consecutive odd integers?
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?
Factorize: $f(a)=4a^4-3a^3-2a^2+3a-2$
Factorize: $(ab+bc+ca)(a+b+c)-abc+(a+b)(b+c)(c+a)$
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) $\mid x_1 - x_2 \mid$