Assume the quartic $x^4-ax^3+bx^2-ax+d=0$ has four real roots $\frac{1}{2}\leq x_1,x_2,x_3,x_4\leq 2.$ Find the maximum possible value of $\frac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}.$

A sequence of real numbers $\{x_n\}$ satisfies the recursion $x_{n+1} = 4x_n - 4x^2_n$, where $n \ge 1$. If $x_{2023} = 0$, compute the number of distinct possible values for $x_1$.

Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows: \[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\] Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence. Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil\equal{}4$, $ \left\lceil 2007\right\rceil\equal{}2007$.

Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

Given non-constant linear functions $p(x), q(x), r(x)$. Prove that at least one of three trinomials $pq+r, pr+q, qr+p$ has a real root. [b]proposed by S. Berlov[/b]

The square polynomial $f(x)=ax^2+bx+c$ is of such a sort, that the equation $f(x)=x$ does not have real roots. Prove that the equation $f(f(x))=0$ does not have real roots also.

The polynomial $P(x) = x^3 - 6x - 2$ has three real roots, $\alpha$, $\beta$, and $\gamma$. Depending on the assignment of the roots, there exist two different quadratics $Q$ such that the graph of $y=Q(x)$ pass through the points $(\alpha,\beta)$, $(\beta,\gamma)$, and $(\gamma,\alpha)$. What is the larger of the two values of $Q(1)$?

For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.

Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer

Let $P(x)$ and $Q(x)$ be (monic) polynomials with real coefficients (the first coefficient being equal to $1$), and $\deg P(x)=\deg Q(x)=10$. Prove that if the equation $P(x)=Q(x)$ has no real solutions, then $ P(x+1)=Q(x-1) $ has a real solution.

In the $xyz$-plane given points $P,\ Q$ on the planes $z=2,\ z=1$ respectively. Let $R$ be the intersection point of the line $PQ$ and the $xy$-plane. (1) Let $P(0,\ 0,\ 2)$. When the point $Q$ moves on the perimeter of the circle with center $(0,\ 0,\ 1)$ , radius 1 on the plane $z=1$, find the equation of the locus of the point $R$. (2) Take 4 points $A(1,\ 1,\ 1) , B(1,-1,\ 1), C(-1,-1,\ 1)$ and $D(-1,\ 1,\ 1)$ on the plane $z=2$. When the point $P$ moves on the perimeter of the circle with center $(0,\ 0,\ 2)$ , radius 1 on the plane $z=2$ and the point $Q$ moves on the perimeter of the square $ABCD$, draw the domain swept by the point $R$ on the $xy$-plane, then find the area.

If $ a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial \[ \left(1\plus{}x\plus{}x^2\right)^{25}\] show that $ a_0\plus{}a_2\plus{}a_4\plus{}\cdots\plus{}a_{50}$ is even.

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

[b]p1.[/b] Catherine's teacher thinks of a number and asks her to subtract $5$ and then multiply the result by $6$. Catherine accidentally switches the numbers by subtracting 6 and multiplying by $5$ to get $30$. If Catherine had not swapped the numbers, what would the correct answer be? [b]p2.[/b] At Acton Boxborough Regional High School, desks are arranged in a rectangular grid-like configuration. In order to maintain proper social distancing, desks are required to be at least 6 feet away from all other desks. Assuming that the size of the desks is negligible, what is the maximum number of desks that can fit in a $25$ feet by $25$ feet classroom? [b]p3.[/b] Joshua hates writing essays for homework, but his teacher Mr. Meesh assigns two essays every $3$ weeks. However, Mr. Meesh favors Joshua, so he allows Joshua to skip one essay out of every $4$ that are assigned. How many essays does Joshua have to write in a $24$-week school year? [b]p4.[/b] Libra likes to read, but she is easily distracted. If a page number is even, she reads the page twice. If a page number is an odd multiple of three, she skips it. Otherwise, she reads the page exactly once. If Libra's book is $405$ pages long, how many pages in total does she read if she starts on page $1$? (Reading the same page twice counts as two pages.) [b]p5.[/b] Let the GDP of an integer be its Greatest Divisor that is Prime. For example, the GDP of $14$ is $7$. Find the largest integer less than $100$ that has a GDP of $3$. [b]p6.[/b] As has been proven by countless scientific papers, the Earth is a flat circle. Bob stands at a point on the Earth such that if he walks in a straight line, the maximum possible distance he can travel before he falls off is $7$ miles, and the minimum possible distance he can travel before he falls off is $3$ miles. Then the Earth's area in square miles is $k\pi$ for some integer $k$. Compute $k$. [b]p7.[/b] Edward has $2$ magical eggs. Every minute, each magical egg that Edward has will double itself. But there's a catch. At the end of every minute, Edward's brother Eliot will come outside and smash one egg on his forehead, causing Edward to lose that egg permanently. For example, starting with $2$ eggs, after one minute there will be $3$ eggs, then $5$, $9$, and so on. After $1$ hour, the number of eggs can be expressed as $a^b + c$ for positive integers $a$, $b$, $c$ where $a > 1$, and $a$ and $c$ are as small as possible. Find $a + b + c$. [b]p8.[/b] Define a sequence of real numbers $a_1$, $a_2$, $a_3$, $..$, $a_{2019}$, $a_{2020}$ with the property that $a_n =\frac{a_{n-1} + a_n + a_{n+1}}{3}$ for all $n = 2$, $3$, $4$, $5$,$...$, $2018$, $2019$. Given that $a_1 = 1$ and $a_{1000} = 1999$, find $a_{2020}$. [b]p9.[/b] In $\vartriangle ABC$ with $AB = 10$ and $AC = 12$, points $D$ and $E$ lie on sides $\overline{AB}$ and $\overline{AC}$, respectively, such that $AD = 4$ and $AE = 5$. If the area of quadrilateral $BCED$ is $40$, find the area of $\vartriangle ADE$. [b]p10.[/b] A positive integer is called powerful if every prime in its prime factorization is raised to a power greater than or equal to $2$. How many positive integers less than 100 are powerful? [b]p11.[/b] Let integers $A,B < 10, 000$ be the populations of Acton and Boxborough, respectively. When $A$ is divided by $B$, the remainder is $1$. When $B$ is divided by $A$, the remainder is $2020$. If the sum of the digits of $A$ is $17$, find the total combined population of Acton and Boxborough. [b]p12.[/b] Let $a_1$, $a_2$, $...$, $a_n$ be an increasing arithmetic sequence of positive integers. Given $a_n - a_1 = 20$ and $a^2_n - a^2_{n-1} = 63$, find the sum of the terms in the arithmetic sequence. [b]p13.[/b] Bob rolls a cubical, an octahedral and a dodecahedral die ($6$, $8$ and $12$ sides respectively) numbered with the integers from $1$ to $6$, $1$ to $8$ and $1$ to $12$ respectively. If the probability that the sum of the numbers on the cubical and octahedral dice equals the number on the dodecahedral die can be written as $\frac{m}{n}$ , where $m, n$ are relatively prime positive integers, compute $n - m$. [b]p14.[/b] Let $\vartriangle ABC$ be inscribed in a circle with center $O$ with $AB = 13$, $BC = 14$, $AC = 15$. Let the foot of the perpendicular from $A$ to BC be $D$ and let $AO$ intersect $BC$ at $E$. Given the length of $DE$ can be expressed as $\frac{m}{n}$ where $m$, $n$ are relatively prime positive integers, find $m + n$. [b]p15.[/b] The set $S$ consists of the first $10$ positive integers. A collection of $10$ not necessarily distinct integers is chosen from $S$ at random. If a particular number is chosen more than once, all but one of its occurrences are removed. Call the set of remaining numbers $A$. Let $\frac{a}{b}$ be the expected value of the number of the elements in $A$, where $a, b$ are relatively prime positive integers. Find the reminder when $a + b$ is divided by $1000$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\] for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$ [b](i)[/b] Prove that $S$ is an infinite set. [b](ii)[/b] Find the least positive integer in $S.$ [b](iii)[/b] If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \] show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]

Suppose $x,y,z$, and $w$ are positive reals such that \[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$. [i]Author: Alex Zhu[/i]

Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $$\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$$

Prove that there is no positive real numbers $a,b,c,d$ such that both of the following equations hold.$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 , \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=32$$.

Let $a$ and $b$ be real numbers, and the polynomial $P(x) =ax + b$ such that $P(2)- P(1)= 3$: Compute the value of $P(5)- P(0)$. [b]A.[/b] $11$ [b]B.[/b] $13$ [b]C.[/b] $15$ [b]D.[/b] $17$ [b]E.[/b] $19$

Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$. Prove that if $a > 0$, then $$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$

When simplified the product $\left(1-\frac13\right)\left(1-\frac14\right)\left(1-\frac15\right)\cdots\left(1-\frac1n\right)$ becomes: $ \textbf{(A)}\ \frac1n \qquad\textbf{(B)}\ \frac2n\qquad\textbf{(C)}\ \frac{2(n-1)}{n}\qquad\textbf{(D)}\ \frac{2}{n(n+1)}\qquad\textbf{(E)}\ \frac{3}{n(n+1)} $

$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?

Prove that if the polynomial $ f(x) = ax^3 + bx^2 + cx + d $ with integer coefficients takes odd values for $ x = 0 $ and $ x = 1 $, then the equation $ f(x) = 0 $ has no integer roots.

Find all $f:R-R$ such that $f(x^2+y+f(y))=2y+f(x)^2$