There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$

A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$. Therefore, each square has exactly $8$ neighbors) What is the maximum possible number of colored squares if$:$ $a) k=6$ $b)k=1$

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [b]Z15.[/b] Let $AOB$ be a quarter circle with center $O$ and radius $4$. Let $\omega_1$ and $\omega_2$ be semicircles inside $AOB$ with diameters $OA$ and $OB$, respectively. Find the area of the region within $AOB$ but outside of $\omega_1$ and $\omega_2$. [u]Set 4[/u] [b]Z16.[/b] Integers $a, b, c$ form a geometric sequence with an integer common ratio. If $c = a + 56$, find $b$. [b]Z17 / D24.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]Z18.[/b] Steven likes arranging his rocks. A mountain formation is where the sequence of rocks to the left of the tallest rock increase in height while the sequence of rocks to the right of the tallest rock decrease in height. If his rocks are $1, 2, . . . , 10$ inches in height, how many mountain formations are possible? For example: the sequences $(1-3-5-6-10-9-8-7-4-2)$ and $(1-2-3-4-5-6-7-8-9-10)$ are considered mountain formations. [b]Z19.[/b] Find the smallest $5$-digit multiple of $11$ whose sum of digits is $15$. [b]Z20.[/b] Two circles, $\omega_1$ and $\omega_2$, have radii of $2$ and $8$, respectively, and are externally tangent at point $P$. Line $\ell$ is tangent to the two circles, intersecting $\omega_1$ at $A$ and $\omega_2$ at $B$. Line $m$ passes through $P$ and is tangent to both circles. If line $m$ intersects line $\ell$ at point $Q$, calculate the length of $P Q$. [u]Set 5[/u] [b]Z21.[/b] Sen picks a random $1$ million digit integer. Each digit of the integer is placed into a list. The probability that the last digit of the integer is strictly greater than twice the median of the digit list is closest to $\frac{1}{a}$, for some integer $a$. What is $a$? [b]Z22.[/b] Let $6$ points be evenly spaced on a circle with center $O$, and let $S$ be a set of $7$ points: the $6$ points on the circle and $O$. How many equilateral polygons (not self-intersecting and not necessarily convex) can be formed using some subset of $S$ as vertices? [b]Z23.[/b] For a positive integer $n$, define $r_n$ recursively as follows: $r_n = r^2_{n-1} + r^2_{n-2} + ... + r^2_0$,where $r_0 = 1$. Find the greatest integer less than $$\frac{r_2}{r^2_1}+\frac{r_3}{r^2_2}+ ...+\frac{r_{2023}}{r^2_{2022}}.$$ [b]Z24.[/b] Arnav starts at $21$ on the number line. Every minute, if he was at $n$, he randomly teleports to $2n^2$, $n^2$, or $\frac{n^2}{4}$ with equal chance. What is the probability that Arnav only ever steps on integers? [b]Z25.[/b] Let $ABCD$ be a rectangle inscribed in circle $\omega$ with $AB = 10$. If $P$ is the intersection of the tangents to $\omega$ at $C$ and $D$, what is the minimum distance from $P$ to $AB$? PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and D.16-30/Z.9-14, 17, 26-30 [url=https://artofproblemsolving.com/community/c3h2916250p26045695]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?

There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?

Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$

We have a $ (n+2)\times n $ rectangle and we’ve divided it into $ n(n+2) \ \ 1\times1 $ squares. $ n(n+2) $ soldiers are standing on the intersection points ($ n+2 $ rows and $ n $ columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a $ n\times(n+2) $ rectangle ($ n $ rows and $ n+2 $ columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left $1$ and $1$ to the right). Prove that $ n $ is even.

We are given $N$ lines ($N > 1$ ) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding $N$ , such that the sum of the integers o n either side of any of the given lines is equal to $0$ . (S . Fomin, Leningrad)

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

IMOC is a small country without any lake. One day, the king decides to divide IMOC into many regions so that each region borders the sea. Prove that the map is $3$-colorable.

How many ways are there to tile a $3 \times 3$ square with $4$ dominoes of size $1 \times 2$ and $1$ domino of size $1 \times 1$? Tilings that can be obtained from each other by rotating the square are considered different. Dominoes of the same size are completely identical

$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).

$n$ ($n \ge 4$) green points are in a data space and no $4$ green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.

Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.

How many subsets of $\{1, 2, ... , 2n\}$ do not contain two numbers with sum $2n+1$?

Let $n\ge 2$ be a positive integer. $S$ is a set of $2n$ airports. For two arbitrary airports $A,B$, if there is an airway from $A$ to $B$, then there is an airway from $B$ to $A$. Suppose that $S$ has only one independent set of $n$ airports. Let the independent set $X$. Prove that there exists an airport $P\in S\setminus X$ which satisfies following condition. [b]Condition[/b] : For two arbitrary distinct airports $A,B\in S\setminus \{P\}$, if there exists a path connecting $A$ and $B$, then there exists a path connecting $A$ and $B$ which does not pass $P$.

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

In Durer’s duck school, there are six rows of doors, as seen on the diagram; both rows are made up of three doors. Dodo duck wishes to enter the school from the street in a way that she uses all six doors exactly once. (On her path, she may go to the street again, or leave the school, so long as she finishes her path in the school.) How many ways can she perform this? [i]Two paths are considered different if Dodo takes the doors in a different order.[/i] [img]https://cdn.artofproblemsolving.com/attachments/5/1/8b722eb2c642e8275928753921fdfbd7495df9.png[/img]

A broken line $A_1A_2 \ldots A_n$ is drawn in a $50 \times 50$ square, so that the distance from any point of the square to the broken line is less than $1$. Prove that its total length is greater than $1248.$

Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?

Let $X =\{1, 2, ... , 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1\le r \le k$, there exist $k$ subsets of $X$ such that the union of any $ r$ of them is equal to $X$ , but the union of any fewer than $r$ of them is not equal to $X$ . Find, with proof, the greatest possible value for $k$.

A $ 100 \times 100$ square floor consisting of $ 10000$ squares is to be tiled by rectangular $ 1 \times 3$ tiles, fitting exactly over three squares of the floor. $ (a)$ If a $ 2 \times 2$ square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles. $ (b)$ If, instead, a $ 2 \times 2$ square is removed from the corner, prove that such a tiling is not possble.

We call the figure shown in the picture consisting of five unit squares a $\emph{plus}$, and each rectangle consisting of two such squares a $\emph{minus}$. Does there exist an odd integer $n$ with the property that a square with side length $n$ can be dissected into pluses and minuses? Justify your answer. [img] https://wiki-images.artofproblemsolving.com//6/6a/18-1-6.png [/img]

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].