In a sports contest, there were $m$ medals awarded on $n$ successive days ($n > 1$). On the first day, one medal and $1/7$ of the remaining $m − 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining
medals were awarded; and so on. On the $n^{th}$ and last day, the remaining $n$ medals were awarded. How many days did the contest last, and how many medals were awarded altogether?
Suppose sequence $\{a_n\}$ satisfies $a_1=0$, $a_2=1$, $a_3=9$, and $S_n^2S_{n-2}=10S_{n-1}^3$ for $n > 3$ where $S_n$ is the sum of the first $n$ terms of this sequence. Find $a_n$ when $n\ge 3$.
Is it possible for a geometric sequence to contain three distinct prime numbers?
Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?
Let $d\ne 0$ be the common difference of an arithmetic sequence $\{a_n\}$, and positive rational number $q < 1$ be the common ratio of a geometric sequence $\{b_n\}$. If $a_1=d$, $b_1=d^2$, and $\frac{a_1^2+a_2^2+a_3^2}{b_1+b_2+b_3}$ is a positive integer, what is the value of $q$?
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$x^2f(x)+f(1-x)=2x-x^4$$
Solve $\{L_n\}$ which is defined as $F_1=1, F_2=3$ and $F_{n+1}=F_{n}+F_{n-1}, (n = 2, 3, 4, \cdots)$
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 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 $b$.
If real number $x$ satisfies $x^4 - 2x^3 -7x^2 + 8x +12\le 0$, find the max value of $|x+\frac{4}{x}|$
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 real numbers $a, b, c$ satisfy $a > 0$, $b>0$, $2c>a+b$, and $c^2>ab$. Prove $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$
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.
Compute the value of $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}+\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
and simplify $$\sqrt[3]{2+\frac{10}{3\sqrt{3}}}\quad\text{and}\quad\sqrt[3]{2-\frac{10}{3\sqrt{3}}}$$
If $abc=1$, solve this equation $$\frac{2ax}{ab+a+1}+\frac{2bx}{bc+b+1}+\frac{2cx}{ca+c+1}=1$$
Let $a$, $b$, and $c$ be three distinct numbers such that $$\frac{a+b}{a-b}=\frac{b+c}{2(b-c)}=\frac{c+a}{3(c-a)}$$
Prove that $8a + 9b + 5c = 0$.
Let $a$, $b$, and $c$ be the lengths of $\triangle{ABC}$'s three sides. Compute the area of $\triangle{ABC}$ if the following relations hold: $$\frac{2a^2}{1+a^2}=b,\qquad \frac{2b^2}{1+b^2}=c,\qquad \frac{2c^2}{1+c^2}=a$$
Let real numbers $x$, $y$, and $z$ satisfy $0 < x, y, z < 1$. Prove $$x(1-y)+y(1-z)+z(1-x)< 1$$
Show that $$\frac{(x+a)(x+b)}{(c-a)(c-b)}+\frac{(x+b)(x+c)}{(a-b)(a-c)}+\frac{(x+c)(x+a)}{(b-c)(b-a)}=1$$ without expanding the left side of the equation.
Find the range of function $f(x)=3^{-|\log_2x|}-4|x-1|$.
Find the minimal value of $y=\sqrt{x^2+2x+5}+\sqrt{x^2-4x+5}$.
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?