Let non-zero real numbers $a, b, c$ satisfy $a+b+c\ne 0$. If the following relations hold $$\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a}$$
Find the value of $$\frac{(a+b)(b+c)(c+a)}{abc}$$
Solve this equation $2x^4 + 3x^3 -16x^2+3x + 2 =0$.
Solve this equation: $(x^2-x-1)^{x+2}=1$.
The sum of two positive integers is $2310$. Show that their product is not divisible by $2310$.
Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.
Suppose $a, b, c$ are all real numbers. If the quadratic polynomial $P(x)=ax^2 + bx + c$ satisfies the condition that $\mid P(x)\mid \le 1$ for all $-1 \le x \le 1$, find the maximum possible value of $b$.
Let $a_1=a_2=1$ and $a_{n}=(a_{n-1}^2+2)/a_{n-2}$ for $n=3, 4, \cdots$. Show that $a_n$ is an integer for $n=3, 4, \cdots$.
Show that every integer $k > 1$ has a multiple which is less than $k^4$ and can be written in base 10 using at most 4 different digits.
Let $a, b, c, p$ be real numbers, with $a, b, c$ not all equal, such that $$a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=p$$ Determine all possible values of $p$ and prove $abc+p=0$.
Show that if a polynomial $P(x)$ satisfies $P(2x^2-1)=(P(x))^2/2$, then it must be a constant.
Suppose $\alpha$ and $\beta$ be two real roots of $x^2-px+q=0$ where $p$ and $q\ne 0$ are two real numbers. Let sequence $\{a_n\}$ satisfies $a_1=p$, $a_2=p^2-q$, and $a_n=pa_{n-1}-qa_{n-2}$ for $n > 2$.
Express $a_n$ using $\alpha$ and $\beta$.
If $p=1$ and $q=\frac{1}{4}$, find the sum of first $n$ terms of $\{a_n\}$.
Compute $$S_n=\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+\cdots+\frac{n+1}{2^n}$$
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 $n\ge 3$.
Find an expression for $x_n$ if sequence $\{x_n\}$ satisfies $x_1=2$, $x_2=3$, and
$$
\left\{
\begin{array}{ccll}
x_{2k+1}&=&x_{2k} +x_{2k-1}&\quad (k\ge 1)\\
x_{2k}&=&x_{2k-1} + 2x_{2k-2}&\quad (k\ge 2)
\end{array}
\right.
$$
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 $S_n$ be the sum of first $n$ terms of an arithmetic sequence. If $S_n=30$ and $S_{2n}=100$, compute $S_{3n}$.
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 $q$?
Let $S_n$ be the sum of the first $n$ terms in geometric sequence $\{a_n\}$. If all $a_n$ are real numbers and $S_{10}=10$, and $S_{30}=70$, compute $S_{40}$.
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 $F_{n+1}=F_{n}+F_{n-1}, (n = 2, 3, 4, \cdots)$