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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)
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 $f(x)=3^{-|\log_2x|}-4|x-1|$.