There is a unit circle that starts out painted white. Every second, you choose uniformly at random an arc of arclength $1$ of the circle and paint it a new color. You use a new color each time, and new paint covers up old paint. Let $c_n$ be the expected number of colors visible after $n$ seconds. Compute $\lim_{n\to \infty} c_n$.

The eight points below are the vertices and the midpoints of the sides of a square. We would like to draw a number of circles through the points, in such a way that each pair of points lie on (at least) one of the circles. Determine the smallest number of circles needed to do this. [asy] unitsize(1 cm); dot((0,0)); dot((1,0)); dot((2,0)); dot((0,1)); dot((2,1)); dot((0,2)); dot((1,2)); dot((2,2)); [/asy]

Let $P$ is a regular $(2n+1)$-gon in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$ , if the line segment $\overline{ES}$ contains no other points that lie on the sides of $P$ except $S$ . We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$ , at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$ , we consider them colorless). Find the largest positive integer $n$ for which such a coloring is possible.

Given a set of n elements, find the largest number of subsets such that no subset is contained in any other

Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows: \[ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}\] where \[ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . \] Prove that the number of elements in $ X $ is a power of 2.

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Let $n \ge 1$ be an integer. Prove that there exists a set $S$ of $n$ positive integers with the following property: if $A$ and $B$ are any two distinct non-empty subsets of $S$, then the averages $\frac{P_{x\in A} x}{|A|}$ and $\frac{P_{x\in B} x}{|B|}$ are two relatively prime composite integers.

A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed [list][b]a)[/b] $1$ centimeter; [b]b)[/b] $1.01$ centimeters; [b]c)[/b] $1.001$ centimeters?[/list]

Consider a checkered board $2m \times 2n$, $m, n \in \mathbb{Z}_{>0}$. A stone is placed on one of the unit squares on the board, this square is different from the upper right square and from the lower left square. A snail goes from the bottom left square and wants to get to the top right square, walking from one square to other adjacent, one square at a time (two squares are adjacent if they share an edge). Determine all the squares the stone can be in so that the snail can complete its path by visiting each square exactly one time, except the square with the stone, which the snail does not visit.

Let $T$ be a tree. Prove that there is a constant $c>0$ (independent of $n$) such that every graph with $n$ vertices that does not contain a subgraph isomorphic to $T$ has at most $cn$ edges. [i]David Yang.[/i]

We consider tilings of a rectangular $m \times n$-board with $1\times2$-tiles. The tiles can be placed either horizontally, or vertically, but they aren't allowed to overlap and to be placed partially outside of the board. All squares on theboard must be covered by a tile. (a) Prove that for every tiling of a $4 \times 2010$-board with $1\times2$-tiles there is a straight line cutting the board into two pieces such that every tile completely lies within one of the pieces. (b) Prove that there exists a tiling of a $5 \times  2010$-board with $1\times 2$-tiles such that there is no straight line cutting the board into two pieces such that every tile completely lies within one of the pieces.

In a company of $n$ persons, each person has no more than $d$ acquaintances, and in that company there exists a group of $k$ persons, $k\ge d$, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than $\left[ \frac{n^2}{4}\right]$.

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?

Every side and diagonal of a regular $12$-gon is colored in one of $12$ given colors. Can this be done in such a way that, for every three colors, there exist three vertices which are connected to each other by segments of these three colors?

We say that a positive integer is a [i]good number[/i] if the digit $2$ appears more often than the digit $3$ and that it is a [i]bad number[/i] if the digit $3$ appears more often than the digit $2$. For example, $2023$ is a good number and $123$ is neither good nor bad. Calculate the difference between the quantity of good numbers and the quantity of bad numbers for integers less than or equal to $2023$.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

Let $A$ and $B$ be two sets of non-negative integers, define $A+B$ as the set of the values obtained when we sum any (one) element of the set $A$ with any (one) element of the set $B$. For instance, if $A=\{2,3\}$ and $B=\{0,1,2,5\}$ so $A+B=\{2,3,4,5,7,8\}$. Determine the least integer $k$ such that there is a pair of sets $A$ and $B$ of non-negative integers with $k$ and $2k$ elements, respectively, and $A+B=\{0,1,2,\dots, 2019,2020\}$

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$. Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.

For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?

Fix an integer $n \ge 1$. On a $m\times n$ chess board, what is the minimum value of $m$ such that $n$ queens can be placed on the chessboard without any two attacking each other? (A queen can attack vertically, horizontally, or diagonally across any distance.)

For which natural numbers $n{}$ it is possible to mark several cells of an $n\times n$ board in such a way that there are an even number of marked cells in all rows and columns, and an odd number on all diagonals whose length is more than one cell? [i]Proposed by S. Berlov[/i]

Two players take turns writing on the board in a row from left to right arbitrary numbers. The player loses, after whose move one or more several digits written in a row form a number divisible by $11$. Which player will win if played correctly?

Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$.