Let $f(x)$ be a function defined on $\mathbb{R}$. If for every real number $x$, the relationships $$f(x+3)\le f(x)+3\quad\text{and}\quad f(x+2)\ge f(x)+2$$ always hold.
1) Show $g(x) = f(x)-x$ is a periodic function.
2) If $f(998)=1002$, compute $f(2000)$
How many equations in the form of $ax^2+bx+c=0$ are there such that $a$, $b$, and $c$ are all single-digit prime numbers and this equation has at least one integer solution?
Real numbers $x, y, z$ satisfy the following equality: $$4(x + y + z) = x^2 + y^2 + z^2$$
Let $M$ be the maximum of $xy + yz + zx$, and let $m$ be the minimum of $xy + yz + zx$. Find $M + 10m$.
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 $c$.
Solve the equation $$\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=(a+1)\sqrt{\frac{x}{x+\sqrt{x}}}$$
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) $\mid x_1 - x_2 \mid$
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 $\frac{mn+4m+1}{n}$?