Let $a_1,a_2,\cdots$, be a sequence with the following properties.
(i) $a_1=1$, and
(ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$.
What is the value of $a_{2^{100}}$?
Solve the following system in integers:
$$
\left\{
\begin{array}{ll}
x_1 + x_2 + \cdots + x_n &= n \\
x_1^2 + x_2^2 + \cdots + x_n^2 &= n \\
\cdots\\
x_1^n + x_2^n + \cdots + x_n^n &= n
\end{array}
\right.
$$
Let $a_1, a_2, \cdots, a_{100}, b_1, b_2, \cdots, b_{100}$ be distinct real numbers. They are used to fill a $100 \times 100$ grids by putting the value of $(a_i + b_j)$ in the cell $(i, j)$ where $1 \le i, j \le 100$. Let $A_i$ be the product of all the numbers in column $i$, and $B_i$ be the product of all the numbers in row $i$. Show that if every $A_i$ equals to 1, then every $B_j$ equals to -1.
Suppose $x, y, z \in \mathbb{R}^+$, such that
\begin{align*}
x^2 + y^2 + xy &= 9\\
y^2 + z^2 + yz &= 16\\
z^2 + x^2 + zx &= 25
\end{align*}
Find the value of $xy+yz+zx$.
Suppose $P(x)$ is a monnic polynomial (meaning the leading coefficient is 1) with 20 roots, each distinct and of the form $\frac{1}{3^k}$ for $k=0, 1, \cdots, 19$. Find the coefficient of $x^{18}$ in $P(x)$.
Find the sum of the reciprocals of all perfect squares whose prime factorization contains only 3, 5, and 7, i.e. $$\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\frac{1}{9}+\frac{1}{81}+\frac{1}{225}+\frac{1}{441}+\frac{1}{625}+\cdots$$
Find the range of the function $$y=x+\sqrt{x^2 -3x+2}$$
For any non-negative real numbers $x$ and $y$, the function $f(x+y^2)=f(x) + 2[f(y)]^2$ always holds, $f(x)\ge 0$, $f(1)\ne 0$. Find the value of $f(2+\sqrt{3})$.
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 $f(2000)$
Which pair contains same functions:
(A) $f(x)=\sqrt{(x-1)^2}$ and $g(x)=x-1$
(B) $f(x)=\sqrt{x^2 -1}$ and $g(x)=\sqrt{x+1}\cdot\sqrt{x-1}$
(C) $f(x)=(\sqrt{x -1})^2$ and $g(x)=\sqrt{(x-1)^2}$
(D) $f(x)\displaystyle\sqrt{\frac{x^2-1}{x+1}}$ and $g(x)=\displaystyle\frac{\sqrt{x^2-1}}{\sqrt{x+1}}$
Select all correct answers.
Let function $f(x)$ is defined as the following: $$ f(x)= \left\{ \begin{array}{ll} x+2 &, \text{if } x \le -1\\ 2x &, \text{if } -1 < x < 2\\ \displaystyle\frac{x^2}{2} &, \text{if } x \ge 2 \end{array} \right. $$ (A) Compute $f(f(f(-\frac{7}{4})))$ (B) If $f(a)=3$, find the value of $a$
Let $a$ and $k$ be two positive integers, and function $f(x)=3x+1$. If $f(x)$'s domain is $\{1, 2, 3, k\}$ and range is $\{4, 7, a^4, a^2 + 3a\}$, find the value of $a$ and $k$.
Which of the following function is the same as $y=\sqrt{-2x^3}$?
(A) $y=x\sqrt{-2x}\qquad$ (B) $y=-x\sqrt{-2x}\qquad$ (C) $y=-\sqrt{2x^3}\qquad$ (D) $y=x^2\sqrt{-2/x}$
Let $f(x)$ be an odd function and $g(x)$ be an even function. If $f(x)+g(x)=\frac{1}{x-1}$, find $f(x)$ and $g(x)$.
If $f\Big(\displaystyle\frac{x+1}{x}\Big)=\displaystyle\frac{x^2+x+1}{x^2}$, find $f(x)$.
If $f(x)$ is an odd function defined on $\mathbb{R}$, compute $f(0)$.
Let function $f(x)$ satisfy $f(a)+f(b)=f(ab)$, and $f(2)=2$ and $f(3)=3$. Compute $f(72)$.
Find the general formula of the sequence defined as $a_1=6$ and $a_n=\frac{1}{2}a_{n-1}+4$.
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?
A function $f$ has its domain equal to the set of integers $\{0, 1, ..., 11\}$, and $f(n)\ge 0$ for all such $n$, and $f$ satisfies: $f(0) = 0$, $f(6) = 1$. If $x \ge 0$, $y\ge 0$, and $x + y\le 11$, then $f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$. Find $f(2)^2 + f(10)^2$.
There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.
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 $M + 10m$.
Given that $x_{n+2} =\frac{20x_{n+1}}{14x_n}$, $x_0 = 25$, $x_1 = 11$, it follows that $$\sum_{n=0}^{\infty}\frac{x_{3n}}{2^n}=\frac{p}{q}$$ for some positive
integers $p, q$ with $GCD(p, q) = 1$. Find $p + q$.
For nonnegative integer $n$, the following are true:
$f(0) = 0$
$f(1) = 1$
$f(n) = f(n-\frac{m(m-1)}{2})-f(\frac{m(m+1)}{2} -n)$ for integer $m$ satisfying $m \ge 2$ and $\frac{m(m-1)}{2} < n \le \frac{m(m+1)}{2}$.
Find the smallest $n$ such that $f(n) = 4$.
Consider all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $$f(f(x)+2x+20)=15$$ Call an integer $n$ $\textit{good}$ if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.