Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$ Find: a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$ b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.

The names of $8$ people are written on slips of paper and placed in a hat. Each of the $8$ people then randomly draw a piece of paper (without replacement). Then, the people are formed into groups satisfying the following requirements: (i) Each person is in the same group as the person who drew his piece of paper. (ii)There are as many groups as possible while still satisfying condition (i). On average, how many groups will there be? (There might be “groups” of only one person.)

Determine the (real) domain of a function $$y=\sqrt{1-\frac{x}{4}|x|+\sqrt{1-\frac{x}{2}|x|\,}\,}-\sqrt{1-\frac{x}{4}|x|-\sqrt{1-\frac{x}{2}|x|\,}\,}$$ and draw its graph.

Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle. [i]Proposed by Jack Edward Smith, UK[/i]

The first four terms in an arithmetic sequence are $ x \plus{} y$, $ x \minus{} y$, $ xy$, and $ x/y$, in that order. What is the fifth term? $ \textbf{(A)}\ \minus{}\frac{15}{8} \qquad \textbf{(B)}\ \minus{}\frac{6}{5} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \frac{27}{20} \qquad \textbf{(E)}\ \frac{123}{40}$

Torus $\mathcal T$ is the surface produced by revolving a circle with radius $3$ around an axis in the plane a distance $6$ from the center of the circle. When a sphere of radius $11$ rests inside $\mathcal T$, it is internally tangent to $\mathcal T$ along a circle with radius $r_{i}$, and when it rests outside $\mathcal T$, it is externally tangent along a circle with radius $r_{o}$. The difference $r_{i}-r_{o}=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

At a dance, Abhinav starts from point $(a, 0)$ and moves along the negative $x$ direction with speed $v_a$, while Pei-Hsin starts from $(0,6)$ and glides in the negative $y$-direction with speed $v_b$. What is the distance of closest approach between the two?

There are 100 blue lines drawn on the plane, among which there are no parallel lines and no three of which pass through one point. The intersection points of the blue lines are marked in red. Could it happen that the distance between any two red dots lying on the same blue line is equal to an integer? [i]From the folklore[/i]

A triangle is divided into nine smaller triangles, where counters with the number zero are placed at each of the ten vertices. A [i]movement[/i] consists of choosing one of the nine triangles and applying the same operation to the three counters of that triangle: adding a unit or subtracting a unit. The figure illustrates a possible [i]movement[/i]. We shall call the integer number n [i]dominant [/i] if it is possible, after a few moves, to obtain a configuration in which the counter numbers are all consecutive and the largest of these numbers is $n$. Determine all [i]dominant [/i] numbers. [img]https://cdn.artofproblemsolving.com/attachments/7/3/731160e6e9a2b3292a31c4555d4adbc7028596.png[/img]

In the coordinate plane, consider points $A = (0, 0)$, $B = (11, 0)$, and $C = (18, 0)$. Line $\ell_A$ has slope 1 and passes through $A$. Line $\ell_B$ is vertical and passes through $B$. Line $\ell_C$ has slope $-1$ and passes through $C$. The three lines $\ell_A$, $\ell_B$, and $\ell_C$ begin rotating clockwise about points $A$, $B$, and $C$, respectively. They rotate at the same angular rate. At any given time, the three lines form a triangle. Determine the largest possible area of such a triangle.

Let $f(x,y)$ be a continuous function on the square $$S=\{(x,y):0\leq x\leq 1, 0\leq y\leq 1\}.$$ For each point $(a,b)$ in the interior of $S$, let $S_{(a,b)}$ be the largest square that is contained in $S$, is centered at $(a,b)$, and has sides parallel to those of $S$. If the double integral $\int \int f(x,y) dx dy$ is zero when taken over each square $S_{(a,b)}$, must $f(x,y)$ be identically zero on $S$?

Let $ABC$ be an acute triangle. Let $D, E,$ and $F$ be the feet of altitudes from $A, B,$ and $C$ to sides $BC, CA,$ and $AB$, respectively, and let $Q$ be the foot of altitude from A to line $EF$ . Given that $AQ = 20, BC = 15,$ and $AD = 24$, compute the perimeter of triangle $DEF.$

All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

Determine the values of $n \in N$ such that a square of side $n$ can be split into a square of side $1$ and five rectangles whose side measures are $10$ distinct natural numbers and all greater than $1$.

$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have $a_ia_{i+1}+1$ is a perfect square(cube). $(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation. $(b)$ Prove that for no natural number $n$ exists a cubic permutation.

Is it possible to cover a $13\times 13$ chessboard with forty-two pieces of dimensions $4\times 1$ such that only the central square of the chessboard remains uncovered?

Thanos establishes $5$ settlements on a remote planet, randomly choosing one of them to stay in, and then he randomly builds a system of roads between these settlements such that each settlement has exactly one outgoing (unidirectional) road to another settlement. Afterwards, the Avengers randomly choose one of the $5$ settlements to teleport to. Then, they (the Avengers) must use the system of roads to travel from one settlement to another. The probability that the Avengers can find Thanos can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. [i]Proposed by Anton Trygub[/i]

If $ y \equal{} \log_{a}{x}$, $ a > 1$, which of the following statements is incorrect? $\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ \textbf{(B)}\ \text{If }x=a,y=1 \qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)} \qquad\\ \textbf{(D)}\ \text{If }0<x<z,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero} \qquad\\ \textbf{(E)}\ \text{Only some of the above statements are correct}$

Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.

The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.

Find the least $n\in N$ such that among any $n$ rays in space sharing a common origin there exist two which form an acute angle.

$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$ $m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.

Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.