Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Determine the highest possible value of: $$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$ where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$. Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.

Determine the natural numbers $n\ge 2$ for which exist $x_1,x_2,...,x_n \in R^*$, such that $$x_1+x_2+...+x_n=\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}=0$$

Solve the following system of equations for real $x,y$ and $z$: \begin{eqnarray*} x &=& \sqrt{2y+3}\\ y &=& \sqrt{2z+3}\\ z &=& \sqrt{2x+3}. \end{eqnarray*}

Prove that there doesn't exist any function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that : $f(f(n-1)=f(n+1)-f(n)$, for every natural $n\geq2$

Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold: $i) $ $f (x+y)\ge f (x)+y $ $ii) $ $f (f (x))\le x $

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Prove that if $a,b$ are positive numbers, then $$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$

Let $a, b, c$ be non-zero real numbers that satisfy $\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b}$. The expression $\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}$ has a maximum value $M$. Find the sum of the numerator and denominator of the reduced form of $M$.

Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\ \frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$

Determine all real numbers $a, b, c, d$ for which $$ab + c + d = 3$$ $$bc + d + a = 5$$ $$cd + a + b = 2$$ $$da + b + c = 6$$

Let $P=(x^4-40x^2+144)(x^3-16x)$. $a)$ Factor $P$ as a product of irreducible polynomials. $b)$ We write down the values of $P(10)$ and $P(91)$. What is the greatest common divisor of the two numbers?

Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.

For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$. Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.

Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.

Solve the system of equations \[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]

The set of all real numbers $ x$ for which \[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x}))) \]is defined is $ \{x|x > c\}$. What is the value of $ c$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$

Solve in integer the equation $\frac{1}{2}(x+y)(y+z)(x+z)+(x+y+z)^{3}=1-xyz$

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.

[b]p1.[/b] How many subsets of $\{D,U,K,E\}$ have an odd number of elements? [b]p2.[/b] Find the coefficient of $x^{12}$ in $(1 + x^2 + x^4 +... + x^{28})(1 + x + x^2 + ...+ x^{14})^2$. [b]p3.[/b] How many $4$-digit numbers have their digits in non-decreasing order from left to right? [b]p4.[/b] A dodecahedron (a polyhedron with $12$ faces, each a regular pentagon) is projected orthogonally onto a plane parallel to one of its faces to form a polygon. Find the measure (in degrees) of the largest interior angle of this polygon. [b]p5.[/b] Justin is back with a $6\times 6$ grid made of $36$ colorless squares. Dr. Kraines wants him to color some squares such that $\bullet$ Each row and column of the grid must have at least one colored square $\bullet$ For each colored square, there must be another colored square on the same row or column What is the minimum number of squares that Justin will have to color? [b]p6.[/b] Inside a circle $C$, we have three equal circles $C_1$, $C_2$, $C_3$, which are pairwise externally tangent to each other and all internally tangent to $C$. What is the ratio of the area of $C_1$ to the area of $C$? [b]p7.[/b] There are $3$ different paths between the Duke Chapel and the Physics building. $6$ students are heading towards the Physics building for a class, so they split into $3$ pairs and each pair takes a separate path from the Chapel. After class, they again split into $3$ pairs and take separate paths back. Find the number of possible scenarios where each student's companion on the way there is different from their companion on the way back. [b]p8.[/b] Let $a_n$ be a sequence that satisfies the recurrence relation $$a_na_{n+2} =\frac{\cos (3a_{n+1})}{\cos (a_{n+1})[2 \cos(2a_{n+1}) - 1]}a_{n+1}$$ with $a_1 = 2$ and $a_2 = 3$. Find the value of $2018a_{2017}$. [b]p9.[/b] Let $f(x)$ be a polynomial with minimum degree, integer coefficients, and leading coefficient of $1$ that satisfies $f(\sqrt7 +\sqrt{13})= 0$. What is the value of $f(10)$? [b]p10.[/b] $1024$ Duke students, indexed $1$ to $1024$, are having a chat. For each $1 \le i \le 1023$, student $i$ claims that student $2^{\lfloor \log_2 i\rfloor +1}$ has a girlfriend. ($\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Given that exactly $201$ people are lying, find the index of the $61$st liar (ordered by index from smallest to largest). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The five real numbers $v,w,x,y,s$ satisfy the system of equations \begin{align*} v&=wx+ys,\\ v^2&=w^2x+y^2s,\\ v^3&=w^3x+y^3s. \end{align*} Show that at least two of them are equal.

4. Let $\mathbb{N}$ denote the strictly positive integers. A function $f$ : $\mathbb{N}$ $\to$ $\mathbb{N}$ has the following properties which hold for all $n \in$ $\mathbb{N}$: a) $f(n)$ < $f(n+1)$; b) $f(f(f(n)))$ = 4$n$ Find $f(2022)$.