Each square in an $n \times n$ table is painted black or white. The routes where two rows meet two columns, called a quartet if the remaining squares are the same color. (a) What is the largest possible number of black squares in a $4 \times 4$ table without quartets? (b) Is it possible to paint a $5 \times 5$ table so that it has no quartets?

Lazim rolls two $24$-sided dice. From the two rolls, Lazim selects the die with the highest number. $N$ is an integer not greater than $24$. What is the largest possible value for $N$ such that there is a more than $50$% chance that the die Lazim selects is larger than or equal to $N$?

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

Vaysha has a board with $999$ consecutive numbers written and $999$ labels of the form [i]"This number is [b]not [/b]divisible by $i$"[/i], for $i \in \{ 2,3, \dots ,1000 \} $. She places each label next to a number on the board, so that each number has exactly one label. For each true statement on the stickers, Vaysha gets a piece of candy. How many pieces of candy can Vaysha guarantee to win, regardless of the numbers written on the board, if she plays optimally?

Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of $5$. If Ana goes first, does either player have a winning strategy?

Three legendary knights are fighting against a multiheaded dragon. Whenever the first knight attacks, he cuts off half of the current number of heads plus one more. Whenever the second knight attacks, he cuts off one third of the current number of heads plus two more. Whenever the third knight attacks, he cuts off one fourth of the current number of heads plus three more. They repeatedly attack in an arbitrary order so that at each step an integer number of heads is being cut off. If all the knights cannot attack as the number of heads would become non-integer, the dragon eats them. Will the knights be able to cut off all the dragon’s heads if it has $41!$ heads? Alexey Zaslavsky

Let $ G$ be an infinite graph such that for any countably infinite vertex set $ A$ there is a vertex $ p$, not in $A$, joined to infinitely many elements of $ A$. Show that $ G$ has a countably infinite vertex set $ A$ such that $ G$ contains uncountably infinitely many vertices $ p$ joined to infinitely many elements of $ A$. [i]P. Erdos, A. Hajnal[/i]

The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$. What are the possible numbers u can get after $99$ consecutive operations of these?

$\definecolor{A}{RGB}{70,255,50}\color{A}\fbox{C6.}$ There are $n$ $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ and $n$ $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ in a club. Some of them are friends with each other. The $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ want to get into a [i]relationship[/i], so some subset of them wants to ask some $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ out for a trip. Because the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ are shy, for a nonempty set $B$ of $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$, they want to make sure that each of the girl they ask out is friend with one of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$. If the number of $\definecolor{A}{RGB}{255,0,255}\color{A}\text{girls}$ they are able to ask out is smaller than the number of the $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ in $B$, then the nonempty set $B$ of those $\definecolor{A}{RGB}{0,0,255}\color{A}\text{boys}$ is called a group of complete losers. Show that for any $0 \le k < 2n$, there exists an arrangement of the [i]friendships[/i] among those $2n$ people so that there are exactly $k$ groups of complete losers. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1737[/color]

What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon? (A Shapovalov)

There are $n \ge 2$ positive integers written on the whiteboard. A move consists of three steps: calculate the least common multiple $N$ of all numbers then choose any number $a$ and replace $a$ by $N/a$ . Prove that, using a finite number of moves, you can always make all the numbers on the whiteboard equal to $ 1$.

Let $S_5$ be the set of permutations of $\{1,2,3,4,5\}$, and let $C$ be the convex hull of the set $$\{(\sigma(1),\sigma(2),\ldots,\sigma(5))\,|\,\sigma\in S_5\}.$$ Then $C$ is a polyhedron. What is the total number of $2$-dimensional faces of $C$?

At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces Anja and Bernd play in turns according to the following rule: [i]When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. [/i] Who has a winning strategy when Anja starts?

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

There are several dominoes on a board such that each domino occupies two adjacent cells and none of the dominoes are adjacent by side or vertex. The bottom left and top right cells of the board are free. A token starts at the bottom left cell and can move to a cell adjacent by side: one step to the right or upwards at each turn. Is it always possible to move from the bottom left to the top right cell without passing through dominoes if the size of the board is a) $100 \times 101$ cells and b) $100 \times 100$ cells? [i]Nikolay Chernyatiev[/i]

[b]p1.[/b] Isosceles trapezoid $ABCD$ has $AB = 2$, $BC = DA =\sqrt{17}$, and $CD = 4$. Point $E$ lies on $\overline{CD}$ such that $\overline{AE}$ splits $ABCD$ into two polygons of equal area. What is $DE$? [b]p2.[/b] At the Berkeley Sandwich Parlor, the famous BMT sandwich consists of up to five ingredients between the bread slices. These ingredients can be either bacon, mayo, or tomato, and ingredients of the same type are indistiguishable. If there must be at least one of each ingredient in the sandwich, and the order in which the ingredients are placed in the sandwich matters, how many possible ways are there to prepare a BMT sandwich? [b]p3.[/b] Three mutually externally tangent circles have radii $2$, $3$, and $3$. A fourth circle, distinct from the other three circles, is tangent to all three other circles. The sum of all possible radii of the fourth circle can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Assume that $ A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $ l$ taken arbitrarily on the plane, assume that projections of $ A_1, A_2, \ldots, A_8$ on the line are $ P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $ P_{i_1}, P_{i_2}, \ldots, P_{i_8}$ according to the direction of line $ l.$ Thus we get one permutation for $ 1, 2, \ldots, 8,$ namely, $ i_1, i_2, \ldots, i_8.$ In the figure, this permutation is $ 2, 1, 8, 3, 7, 4, 6, 5.$ Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $ N_8 \equal{} N(A_1, A_2, \ldots, A_8).$ Find the maximal value of $ N_8.$

Alice and Bob play the following game. To start, Alice arranges the numbers $1,2,\ldots,n$ in some order in a row and then Bob chooses one of the numbers and places a pebble on it. A player's [i]turn[/i] consists of picking up and placing the pebble on an adjacent number under the restriction that the pebble can be placed on the number $k$ at most $k$ times. The two players alternate taking turns beginning with Alice. The first player who cannot make a move loses. For each positive integer $n$, determine who has a winning strategy.

We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed the center of rectangle(Intersection of diagonals.)

In the playboard shown beside, players $A$ and $B$ alternately fill the empty cells by integers, player $A$ starting. In each step the empty cell and the integer can be chosen arbitrarily. Show that player $A$ can always achieve that all the equalities hold after the last step. [img]https://cdn.artofproblemsolving.com/attachments/c/0/524195b1a8ab8457b72005a162f8124c2b1bd2.png[/img]

Each citizen in a town knows at least $ 30$% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least $ 50$% of the citizen can vote.

There are $5$ people at a party. For each pair of people, there is a $1/2$ chance they are friends, independent of all other pairs. Find the expected number of pairs of people who have a mutual friend, but are not friends themselves. [i]Proposed by Patrick Xue[/i]

The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves? [i](Proposed by Avan Lim Zenn Ee)[/i]

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i, j)$ and $(j, i)$ do not [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.