Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$ If $N=100a+b$, where $a,b$ are positive integers less than $100$, find $a+b.$

Let $\mathcal{A}_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$ with $x_i \in \{0, 1, 2\}$. A triple $x, y, z$ of distinct elements of $\mathcal{A}_n$ is called [i]good[/i] if there is some $i$ such that $\{x_i, y_i, z_i\} = \{0, 1, 2\}$. A subset $A$ of $\mathcal{A}_n$ is called [i]good[/i] if every three distinct elements of $A$ form a good triple. Prove that every good subset of $\mathcal{A}_n$ has at most $2(\frac{3}{2})^n$ elements.

An L-shaped figure composed of $4$ unit squares (such as shown in the picture) we call L-dominoes. [img]https://cdn.artofproblemsolving.com/attachments/b/2/064b7c7de496f981cd937cbb7392efc1066420.png[/img] Determine the maximum number of L-dominoes that can be placed on a board of dimensions $n \times n$, where $n$ is natural number, so that no two dominoes overlap and it is possible get from the upper left to the lower right corner of the board by moving only across those squares that are not covered by dominoes. (By moving, we move from someone of the square on it the neighboring square, i.e. the square with which it shares the page). Note: L-Dominoes can be rotated as well as flipped, giving an symmetrical figure wrt axis compared to the one shown in the picture.

Nina and Tadashi play the following game. Initially, a triple $(a, b, c)$ of nonnegative integers with $a+b+c=2021$ is written on a blackboard. Nina and Tadashi then take moves in turn, with Nina first. A player making a move chooses a positive integer $k$ and one of the three entries on the board; then the player increases the chosen entry by $k$ and decreases the other two entries by $k$. A player loses if, on their turn, some entry on the board becomes negative. Find the number of initial triples $(a, b, c)$ for which Tadashi has a winning strategy.

Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.

Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?

The following solitaire game is played on an $m\times n$ rectangular board, $m,n\ge 2$, divided into unit squares. First, a rook is placed on some square. At each move, the rook can be moved an arbitrary number of squares horizontally or vertically, with the extra condition that each move has to be made in the $90^{\circ}$ clockwise direction compared to the previous one (e.g. after a move to the left, the next one has to be done upwards, the next one to the right etc). For which values of $m$ and $n$ is it possible that the rook visits every square of the board exactly once and returns to the first square? (The rook is considered to visit only those squares it stops on, and not the ones it steps over.)

A floor has $2 \times 11$ dimension, and this floor covering with $1 \times 2$ rectangles. (No two rectangles overlap). How many cases we done this job?

Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?

A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)

The positive numbers $a_1, a_2, \ldots , a_{2024}$ are placed on a circle clockwise in this order. Let $A_i$ be the arithmetic mean of the number $a_i$ and one or several following it clockwise. Prove that the largest of the numbers $A_1, A_2, \ldots , A_{2024}$ is not less than the arithmetic mean of all numbers $a_1, a_2, \ldots , a_{2024}$.

[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$? [b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight? [b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together? [b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible. [b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer? [b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time? [b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$. [b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)? [b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. [b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$? [b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?

Anja and Bernd take turns in removing stones from a heap, initially consisting of $n$ stones ($n \ge 2$). Anja begins, removing at least one but not all the stones. Afterwards, in each turn the player has to remove at least one stone and at most as many stones as removed in the preceding move. The player removing the last stone wins. Depending on the value of $n$, which player can ensure a win?

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ ([i]Dan Schwarz[/i])

There are $30$ bags and there are $100$ similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most $999$ grams in each weighing. Using this scale, we want to find the weight of coins of each bag. [b](a)[/b] Show that this operation is possible by $10$ times of weighing, and [b](b)[/b] It's not possible by $9$ times of weighing.

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely: Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: [b](1)[/b] Choose $k+1$ boxes; [b](2)[/b] Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. [b](3)[/b] If one of the boxes is left empty, the game ends; otherwise, go to the next step. [i]Proposed by Demetres Christofides, Cyprus[/i]

There are 2000 cities in Graphland; some of them are connected by roads. For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?

Abolf is on the second step of a stairway to heaven in every step of this stairway except the first one which is the hell there is a devil who is either a human, an elf or a demon and tempts Abolf. The devil in the second step is Satan himself as one of three forms. Whenever an elf or a demon tries to tempt Abolf he resists and walks one step up but when a human tempts Abolf he is deceived and hence he walks one step down. However if Abolf is deceived by Satan for the first time he resists and does not fall down to hell but the second time he falls down to eternal hell. Every time a devil makes a temptation it changes its form from a human, an elf, a demon to an elf, a demon, a human respectively. Prove that Abolf passes each step after some time. [i]Proposed by Yaser Ahmadi Fouladi[/i]

For a given integer $n\ge 2$, let $a_0,a_1,\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\{ a_i+a_j \mid 0\le i \le j \le n \}$.

In each square of a $2012\times 2012$ board there's a person. People are either honest, who always tell the truth, or liars, who always lie. At a given moment, each person makes the same statement: "In my row there are the same number of liars as in my column." Determine the minimum number of honest people that can be on the board.

A square of side $5$ is divided into $25$ unit squares. Let $A$ be the set of the $16$ interior points of the initial square which are vertices of the unit squares. What is the largest number of points of $A$ no three of which form an isosceles right triangle?

Let $n$ be a positive integer. The numbers $1,2,3,\dots, 4n$ are written in a board. Olive wants to make some "couples" of numbers, such that the product of the numbers in each couple is a perfect square. Each number is in, at most, one couple and the two numbers in each couple are distincts. Determine, for each positive integer $n$, the maximum number of couples that Olive can write.

Alice has a deck of $36$ cards, $4$ suits of $9$ cards each. She picks any $18$ cards and gives the rest to Bob. Now each turn Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bob gains a point. What is the maximum number of points he can guarantee regardless of Alice’s actions? Mikhail Evdokimov

A dot-trapezium consists of several rows of dots such that each row contains one more dot than the row immediately above (apart from the top row). For example here is a dot-trapezium consisting of $15$ dots, having $3$ rows and $4$ dots in the top row. [asy] //wonderfully scuffed asymptote code , please don't laugh at me. constructed from the diagram at https://www.mathsolympiad.org.nz/competitions/nzmo/problems/nzmo1_2024.pdf //top row dot((.05,.1)); dot((-.05,.1)); dot((-.15,.1)); dot((.15,.1)); //middle row dot((0,0)); dot((.1,0)); dot((-.1,0)); dot((.2,0)); dot((-.2,0)); //bottom row dot((.05,-.1)); dot((-.05,-.1)); dot((-.15,-.1)); dot((.15,-.1)); dot((.25,-.1)); dot((-.25,-.1)); [/asy] A positive integer $n$ is called a trapezium-number if there exists a dot-trapezium consisting of exactly $n$ dots, with at least two rows and at least two dots in the top row. How many trapezium-numbers are there less than $100$?