Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that
$$f(yf(x)-x)=f(x)f(y)+2x$$
for all $x,\ y\in{\mathbb{R}}$.
For each integer $a_0 >$ 1, define the sequence $a_0, a_1, a_2, \cdots$ by:
$$
a_{n+1} =
\left\{
\begin{array}{ll}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}\\
a_n + 3 & \text{otherwise}
\end{array}
\right.
$$
For all $n \ge 0$. Determine all values of $a_0$ for which there is a number $A$ such that $a_n = A$ for infinitely many values of $n$.
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that, for all real numbers $x$ and $y$,
$$f (f(x)f(y)) + f(x + y) = f(xy)$$
An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is 1. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0$, $a_1$, $\cdots$, an such that, for each $(x, y)$ in $S$, we have:
$$a_0x^n + a_1x^{n-1}y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1$$
Let $P(x)$ be a monic cubic polynomial. The lines $y = 0$ and $y = m$ intersect $P(x)$ at points $A$, $C$, $E$ and $B$, $D$, $F$ from left to right for a positive real number $m$. If $AB = \sqrt{7}$, $CD = \sqrt{15}$, and $EF = \sqrt{10}$, what is the value of $m$?
Show that $$x+n=\sqrt{n^2 + x\sqrt{n^2+(x+n)\sqrt{n^2+(x+2n)\sqrt{\cdots}}}}$$
(Sophie Germain's Identity) Prove $a^4 + 4b^4 = (a^2+2b^2-2ab)(a^2+2b^2+2ab)$.
Find the value of $$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$$
Let $\{x_n\}$ and $\{y_n\}$ be two real number sequences which are defined as follow:
$$x_1=y_1=\sqrt{3},\quad x_{n+1}=x_n +\sqrt{1+x_n^2},\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$
for all $n\ge 1$. Prove that $2 < x_ny_n < 3$ for all $n>1$.
Solve this equation $$2\sqrt{2}x^2 + x -\sqrt{1-x ^2}-\sqrt{2}=0$$
John uses the equation method to evaluate the following expression:$$S=1-1+1-1+1-\cdots$$ and get $$S=1-S \implies \boxed{S=\frac{1}{2}}$$
However, $S$ clearly cannot be a fraction. Can you point out what is wrong here?
Find the remainder when $x^{2017}$ is divided by $(x+1)^2$.
Let $f(x)=2016x - 2015$. Solve this equation $$\underbrace{f(f(f(\cdots f(x))))}_{2017\text{ iterations}}=f(x)$$
Let integers $u$ and $v$ be two integral roots to the quadratic equation $x^2 + bx+c=0$ where $b+c=298$. If $u < v$, find the smallest possible value of $v-u$.
The Fibonacci sequence $(F_n)_{n\ge 0}$ is defined by the recurrence relation $F_{n+2}=F_{n+1}+F_{n}$ with $F_{0}=0$ and $F_{1}=1$. Prove that for any $m$, $n \in \mathbb{N}$, we have
$$F_{m+n+1}=F_{m+1}F_{n+1}+F_{m}F_{n}.$$
Deduce from here that $F_{2n+1}=F^2_{n+1}+F^2_{n}$ for any $n \in \mathbb{N}$
Let $f$ be a function from $\mathbb{N}$ to $\mathbb{N}$ such that
(i) $f(1)=0$
(ii) $f(2n)=2f(n)+1)$
(iii) $f(2x+1)=2f(n)$
Find the least value of $n$ such that $f(n)=2016$.
The sum and product of two numbers are equal to $y$. For which values of $y$ are these two numbers real?
Let $m$ and $n$ be the roots of $P(x)=ax^2+bx+c$. Find the coefficients of the quadratic polynomial whose roots are $m^2-n$ and $n^2-m$.
The roots of $x^2+ax+b+1$ are positive integers. Show that $a^2+b^2$ is not a prime number.
Let $\alpha$ and $\beta$ be the roots of $x^2+px+1$, and let $\gamma$ and $\sigma$ be the roots of $x^2+qx+1$.
Show
$$(\alpha - \gamma)(\beta-\gamma)(\alpha+\sigma)(\beta+\sigma) = q^2 - p^2$$
Let $a$, $b$, $c$ be distinct real numbers. Show that there is a real number $x$ such that
$$x^2+2(a+b+c)x+3(ab+bc+ac)$$
is negative.
Consider the quadratic equation $ax^2-bx+c=0$ where $a$, $b$, $c$ are real numbers and $a \ne 0$. Find the values of $a$, $b$, $c$ such that $a$ and $b$ are the roots of the equation and $c$ is it's discriminant.
Let $b \ge 0$ be a real number. The product of the four real roots of the equations $x^2+2bx+c=0$ and $x^2+2cx+b=0$ is equal to $1$. Find the values of $b$ and $c$.
Solve the equation
$$x^4-97x^3+2012x^2-97x+1=0$$
Show that if $a$, $b$, $c$ are the lengths of the sides of a triangle, then the equation
$$b^2x^2+(b^2+c^2-a^2)x + c^2=0$$
does not have any real roots.