If the middle term of three consecutive integers is a perfect square, then the product of these three numbers is called a $\textit{beautiful}$ number. What is the greatest common divisor of all the $\textit{beautiful}$ numbers?
Find the largest 7-digit integer such that all its 3-digit subpart is either a multiple of 11 or multiple of 13.
Find a $4$-digit square number $x$ such that if every digit of $x$ is increased by 1, the new number is still a perfect square.
If $a+b=\sqrt{5}$, compute $\frac{a^2 -a^2b^2 + b^2 +2ab}{a+ab+b}$.
What is the units digit of $-1\times 2008 + 2 \times 2007 - 3\times 2006 + 4\times 2005 +\cdots-1003\times 1006 + 1004 \times 1005$?
If $-1 < a < b < 0$, then which relationship below holds?
$(A)\quad a < a^3 < ab^2 < ab \qquad (B)\quad a < ab^2 < ab < a^3 \qquad (C)\quad a< ab < ab^2 < a^3 \qquad (D) a^3 < ab^2 < a < ab$
Let point $A$ and $B$ represent real numbers $a$ and $b$, respectively. If $A$ and $B$ lay on different sides of the origin $O$, and $|a - b| = 2016$, $AO = 2BO$, what is the value of $a+b$?
If $a+b=\sqrt{5}$, compute the value of $\frac{a^2 - a^2b^2 + b^2 +2ab}{a+ab+b}+ab$.
If $AE=DE=5$, $AB=CD$, $BC=4$, $\angle{E}=60^\circ$, $\angle{A}=\angle{D}=90^\circ$, then what is the area of the pentagon $ABCDE$?
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)
How many solutions does the following system have?
$$
\left\{
\begin{array}{ll}
\lfloor x \rfloor + 2y &= 1\\
\lfloor y \rfloor + x &=2
\end{array}
\right.
$$
Where $\lfloor x \rfloor$ and $\lfloor y \rfloor$ denote the largest integers not exceeding $x$ and $y$, respectively.
Let $X$ be the integer part of $\left(3+\sqrt{7}\right)^n$ where $n$ is a positive integer. Show that $X$ must be odd.
Is it possible to arrange these numbers, $1, 1, 2, 2, 3, 3, \cdots, 1986, 1986$ to form a sequence for such there is $1$ number between two $1$'s, $2$ numbers between two $2$'s, $\cdots$, $1986$ numbers between two 1986's?
Let even function $f(x)$ and odd function $g(x)$ satisfy the relationship of $f(x)+g(x)=\sqrt{1+x+x^2}$. Find $f(3)$.
Let $f\Big(\dfrac{1}{x}\Big)=\dfrac{1}{x^2+1}$. Compute $$f\Big(\dfrac{1}{2013}\Big)+f\Big(\dfrac{1}{2012}\Big)+f\Big(\dfrac{1}{2011}\Big)+\cdots +f\Big(\dfrac{1}{2}\Big)+f(1)+f(2)+\cdots +f(2011)+f(2012)+f(2013)$$
Let $f(x)=x^{-\frac{k^2}{2}+\frac{3}{2}k+1}$ be an odd function where $k$ is an integer. If $f(x)$ is monotonically increasing when $x\in(0,+\infty)$, find all the possible values of $k$.
Let $G$ be the centroid of $\triangle{ABC}$. Points $M$ and $N$ are on side $AB$ and $AC$, respectively such that $\overline{AM} = m\cdot\overline{AB}$ and $\overline{AN} = n\cdot\overline{AC}$ where $m$ and $n$ are two positive real numbers. Find the minimal value of $mn$.
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)
An infinite number of equilateral triangles are constructed as shown on the right. Each inner triangle is inscribed in its immediate outsider and is shifted by a constant angle $\beta$.
If the area of the biggest triangle equals to the sum of areas of all the other triangles, find the value of $\beta$ in terms of degrees.
![](data:image/png;base64,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)
Let $BD$ be a median in $\triangle{ABC}$. If $AB=\frac{4\sqrt{6}}{3}$, $\cos{B}=\dfrac{\sqrt{6}}{6}$, and $BD=\sqrt{5}$, find the length of $BC$ and the value of $\sin{A}$.
![](data:image/png;base64,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)
Suppose function $f(x)=\frac{1+x}{1-x}$. Evaluate $$f\Big(\frac{1}{2}\Big)\cdot f\Big(\frac{1}{4}\Big)\cdot f\Big(\frac{1}{6}\Big)\cdots f\Big(\frac{1}{2014}\Big)$$
Let the sum of first $n$ terms of arithmetic sequence $\{a_n\}$ be $S_n$, and the sum of first $n$ terms of arithmetic sequence $\{b_n\}$ be $T_n$. If $\frac{S_n}{T_n}=\frac{2n}{3n+7}$, compute the value of $\frac{a_8}{b_6}$.
Suppose every term in the sequence
$$1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, \cdots$$
is either $1$ or $2$. If there are exactly $(2k-1)$ twos between the $k^{th}$ one and the $(k+1)^{th}$ one, find the sum of its first $2014$ terms.
Let $c_1, c_2, c_3, \cdots$ be a series of concentric circles whose radii form a geometric sequence with common ratio as $r$. Suppose the areas of rings which are formed by two adjacent circles are $S_1, S_2, S_3, \cdots$. Which statement below is correct regarding the sequence $\{S_n\}$?
A) It is not a geometric sequence
B) It is a geometric sequence and its common ratio is $r$
C) It is a geometric sequence and its common ratio is $r^2$
D) It is a geometric sequence and its common ratio is $r^2-1$
Given the sequence $\{a_n\}$ satisfies $a_n+a_m=a_{n+m}$ for any positive integers $n$ and $m$. Suppose $a_1=\frac{1}{2013}$. Find the sum of its first $2013$ terms.
If real number $x$ satisfies $x^4 - 2x^3 -7x^2 + 8x +12\le 0$, find the max value of $|x+\frac{4}{x}|$
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.