Expanding $$\Big(\sqrt{x}+\frac{1}{2\sqrt[4]{x}}\Big)^n$$
and arranging all the terms in descending order of $x$'s power, if the coefficients of the first three terms form an arithmetic sequence, how many terms with integer power of $x$ are there?
Suppose sequence $\{F_n\}$ is defined as $$F_n=\frac{1}{\sqrt{5}}\Big[\Big(\frac{1+\sqrt{5}}{2}\Big)^n-\Big(\frac{1-\sqrt{5}}{2}\Big)^n\Big]$$
for all $n\in\mathbb{N}$. Let $$S_n=C_n^1\cdot F_1 + C_n^2\cdot F_2+\cdots +C_n^n\cdot F_n.$$
Find all positive integer $n$ such that $S_n$ is divisible by 8.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+f(1-x)=2x-x^4$$
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that the Cauchy equation $$f(x+y)=f(x)+f(y)$$ holds for all $x, q\in\mathbb{Q}$.
Solve $\{L_n\}$ which is defined as $F_1=1, F_2=3$ and $F_{n+1}=F_{n}+F_{n-1}, (n = 2, 3, 4, \cdots)$
Let real numbers $a, b, c, d$ satisfy
$$
\left\{
\begin{array}{ccl}
ax+by&=3\\
ax^2+by^2&=7\\
ax^3+by^3&=16\\
ax^4 + by^4 &=42
\end{array}
\right.
$$
Find $ax^5+by^5$.
Find the range of function $y=x+\sqrt{x^2 -3x+2}$.
Solve $$\Big|\frac{1}{\log_{\frac{1}{2}}x+2}\Big|> \frac{3}{2}$$
If for any non-negative real numbers $x$ and $y$, function $f(x)$ satisfies the properties that $f(x)\ge 0$, $f(1)\ne 0$, and $f(x+y^2)=f(x)+2f^2(y)$ , compute the value of $f(2+\sqrt{3})$.
If the minimal and maximum values of function $$f(x)=-\frac{1}{2}x^2 + \frac{13}{2}$$ in the domain $[a, b]$ are $2a$ and $2b$, respectively, determine the values of $a$ and $b$.
Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?
For any real numbers $x$ and $y$, the following holds $$[f(x+y)]^2 = [f(x)]^2 + [f(y)]^2$$
Find the exact form of $f(x)$.
Let $f(x)$ be a polynomial with respect to $x$ and $$f(x+1)+f(x-1)=2x^2-4x$$ Find $f(x)$.
Find the function $f(x)$ such that $f(0)=1$, $f(\frac{\pi}{2})=2$, and for any $x, y\in\mathbb{R}$, $$f(x+y)+f(x-y)=2f(x)\cos y$$
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any $m, n\in\mathbb{N}$, $$f(m+n)=f(m)+f(n)+mn$$
Determine $f(n)$.
Let the domain of function $f(n)$ be $\mathbb{N}$, $f(1)=1$, and for any integer $n \ge 2$, $$f(n)=f(n-1) + 2^{n-1}$$
Determine $f(n)$.
Let real numbers $a$, $b$, and $c$ satisfy $a+b+c=2$ and $abc=4$. Find
the minimal value of the largest among $a$, $b$, and $c$.
the minimal value of $\mid a\mid +\mid b \mid +\mid c \mid$.
If $a\ne 0$ and $\frac{1}{4}(b-c)^2=(a-b)(c-a)$, compute $\frac{b+c}{a}$.
If all roots of the equation $$x^4-16x^3+(81-2a)x^2 +(16a-142)x+(a^2-21a+68)=0$$ are integers, find the value of $a$ and solve this equation.
Let real numbers $a, b, c$ satisfy $a > 0$, $b>0$, $2c>a+b$, and $c^2>ab$. Prove $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$
Suppose the graph of $f(x)=x^4 + ax^3 + bx^2 + cd + d$, where $a$, $b$, $c$, $d$ are all real constants, passes through three points $A \big(2,\frac{1}{2}\big)$, $B \big(3, \frac{1}{3}\big)$, and $C \big(4, \frac{1}{4}\big)$. Find the value of $f(1) + f(5)$.
Find a quadratic polynomial $f(x)=x^2 + mx +n$ such that $$f(a)=bc,\quad f(b) = ca,\quad f(c) = ab$$ where $a$, $b$, $c$ are three distinct real numbers.
If all coefficients of the polynomial $$f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_3x^3+x^2+x+1=0$$ are real numbers, prove that its roots cannot be all real.
Compute the value of $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}+\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
and simplify $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
Prove there cannot exist a $998$-degree polynomial with real number coefficients $P(x)$ such that $$[P(x)]^2-1=P(x^2+1)$$ holds for any $x\in\mathbb{C}$.