If $y=a+\frac{b}{x}$, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals: $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11 $

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

Let $a_1, a_2, \dots$ and $b_1, b_2, \dots$ be sequences of real numbers for which $a_1 > b_1$ and \begin{align*} a_{n+1} &= a_n^2 - 2b_n\\ b_{n+1} &= b_n^2 - 2a_n \end{align*} for all positive integers $n$. Prove that $a_1, a_2, \dots$ is eventually increasing (that is, there exists a positive integer $N$ for which $a_k < a_{k+1}$ for all $k > N$). [i]Holden Mui[/i]

Suppose that $x_1, x_2, x_3, . . . x_n$ are real numbers between $0$ and $ 1$ with sum $s$. Prove that $$\prod_{i=1}^{n} \frac{x_i}{s + 1 - x_i} + \prod_{i=1}^{n} (1 - x_i) \le 1.$$

If the ratio $$\frac{17m+43n}{m-n}$$ is an integer where $m$ and $n$ positive integers, let's call $(m,n)$ is a special pair. How many numbers can be selected from $1,2,..., 2021$, any two of which do not form a special pair?

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

For each pair of even natural numbers $ k $, $ m $determine all real numbers $ x $that satisfy the equation $$ (\sin x)^k + (\cos x)^{-m} = (\cos x)^k + (\sin x)^{-m}$$

Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.

Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

Let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{\left(n-1\right)a_{n-1}+\left(n-2\right)a_{n-2}+...+2a_{2}+1a_{1}}{\left(n-1\right)+\left(n-2\right)+...+2+1}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. [i]Generalization.[/i] Let $\left(b_{1},\ b_{2},\ b_{3},\ ...\right)$ be a monotonically increasing sequence of positive reals, and let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{b_{n-1}a_{n-1}+b_{n-2}a_{n-2}+...+b_{2}a_{2}+b_{1}a_{1}}{b_{n-1}+b_{n-2}+...+b_{2}+b_{1}}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. darij

$a,b,c$ are positive real numbers which satisfy $5abc>a^3+b^3+c^3$. Prove that $a,b,c$ can form a triangle.

$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

[b]p1.[/b] Consider the equation $$x_1x_2 + x_2x_3 + x_3x_4 + · · · + x_{n-1}x_n + x_nx_1 = 0$$ where $x_i \in \{1,-1\}$ for $i = 1, 2, . . . , n$. (a) Show that if the equation has a solution, then $n$ is even. (b) Suppose $n$ is divisible by $4$. Show that the equation has a solution. (c) Show that if the equation has a solution, then $n$ is divisible by $4$. [b]p2.[/b] (a) Find a polynomial $f(x)$ with integer coefficients and two distinct integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. (b) Let $f(x)$ be a polynomial with integer coefficients and $a$, $b$, and $c$ be three integers. Suppose $f(a) = b$, $f(b) = c$, and $f(c) = a$. Show that $a = b = c$. [b]p3.[/b] (a) Consider the triangle with vertices $M$ $(0, 2n + 1)$, $S$ $(1, 0)$, and $U \left(0, \frac{1}{2n^2}\right)$, where $n$ is a positive integer. If $\theta = \angle MSU$, prove that $\tan \theta = 2n - 1$. (b) Find positive integers $a$ and $b$ that satisfy the following equation. $$arctan \frac18 = arctan \,\,a - arctan \,\, b$$ (c) Determine the exact value of the following infinite sum. $$arctan \frac12 + arctan \frac18 + arctan \frac{1}{18} + arctan \frac{1}{32}+ ... + arctan \frac{1}{2n^2}+ ...$$ [b]p4.[/b] (a) Prove: $(55 + 12\sqrt{21})^{1/3} +(55 - 12\sqrt{21})^{1/3}= 5$. (b) Completely factor $x^8 + x^6 + x^4 + x^2 + 1$ into polynomials with integer coefficients, and explain why your factorization is complete. [b]p5.[/b] In this problem, we simulate a hula hoop as it gyrates about your waist. We model this situation by representing the hoop with a rotating a circle of radius $2$ initially centered at $(-1, 0)$, and representing your waist with a fixed circle of radius $1$ centered at the origin. Suppose we mark the point on the hoop that initially touches the fixed circle with a black dot (see the left figure). As the circle of radius $2$ rotates, this dot will trace out a curve in the plane (see the right figure). Let $\theta$ be the angle between the positive x-axis and the ray that starts at the origin and goes through the point where the fixed circle and circle of radius $2$ touch. Determine formulas for the coordinates of the position of the dot, as functions $x(\theta)$ and $y(\theta)$. The left figure shows the situation when $\theta = 0$ and the right figure shows the situation when $\theta = 2pi/3$. [img]https://cdn.artofproblemsolving.com/attachments/8/6/d15136872118b8e14c8f382bc21b41a8c90c66.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

Find all real solutions of the system of equations $$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\ (x_4 + x_5 + x_1)^5 = 3x_2\\ (x_5 + x _1 + x_2)^5 = 3x_3\\ (x_1 + x_2 + x_3)^5 = 3x_4\\ (x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$ (L. Tumescu , Romania)

Let $n\ge 2$ be a positive integer. Let $x_1, x_2, \dots, x_n$ be non-zero real numbers satisfying the equation \[\left(x_1+\frac{1}{x_2}\right)\left(x_2+\frac{1}{x_3}\right)\dots\left(x_n+\frac{1}{x_1}\right)=\left(x_1^2+\frac{1}{x_2^2}\right)\left(x_2^2+\frac{1}{x_3^2}\right)\dots\left(x_n^2+\frac{1}{x_1^2}\right).\] Find all possible values of $x_1, x_2, \dots, x_n$. [i]Proposed by Victor Domínguez[/i]

For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.

Let a sequence $\left\{ {{x_n}} \right\}$ defined by: \[\left\{ \begin{array}{l} {x_0} = - 2 \\ {x_n} = \frac{{1 - \sqrt {1 - 4{x_{n - 1}}} }}{2},\forall n \ge 1 \\ \end{array} \right.\] Denote $u_n=n.x_n$ and ${v_n} = \prod\limits_{i = 0}^n {\left( {1 + x_i^2} \right)} $. Prove that $\left\{ {{u_n}} \right\}$, $\left\{ {{v_n}} \right\}$ have finite limit.

Let $f$ be a polynomial function such that, for all real $x$, \[f(x^2 + 1) = x^4 + 5x^2 + 3.\] For all real $x$, $f(x^2-1)$ is $ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.

Prova that for any real $a$, expresssion $A=(a-1)(a-3)(a-4)(a-6)+10$ is always positive. What is the minimum value that expression $A$ can take and for which values of $a$?