Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

Let $n$ be a natural number. A regular hexagon with edge length $n$ gets split into equilateral exploded triangles whose edges are $1$ in length and parallel to one side of the hexagon. Find the number of regular hexagons, the angles of which are all angles of these triangles are.

Alex starts at the origin $O$ of a hexagonal lattice. Every second, he moves to one of the six vertices adjacent to the vertex he is currently at. If he ends up at $X$ after $2018$ moves, then let $p$ be the probability that the shortest walk from $O$ to $X$ (where a valid move is from a vertex to an adjacent vertex) has length $2018$. Then $p$ can be expressed as $\tfrac{a^m-b}{c^n}$, where $a$, $b$, and $c$ are positive integers less than $10$; $a$ and $c$ are not perfect squares; and $m$ and $n$ are positive integers less than $10000$. Find $a+b+c+m+n$.

Fix an integer $n \ge 2$. A fairy chess piece [i]leopard [/i] may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3n \times 3n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made. Dmitry Khramtsov, Russia

we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down) for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$. for example we have such table for $n=5$ 1 0 3 3 4 1 3 2 1 1 0 1 0 1 0 2 1 0 0 0 1 0 0 0 0 prove that for $n>5$ there is no $selfish$ table

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.

Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.) [i](Poland) Marek Cygan[/i]

On the Cartesian plane, all points with both integer coordinates are painted blue. Two blue points are said to be [i]mutually visible[/i] if the line segment that connects them has no other blue points . Prove that there is a set of $ 2019$ blue points that are mutually visible two by two. [hide=official wording]No plano cartesiano, todos os pontos com ambas coordenadas inteiras são pintados de azul. Dois pontos azuis são ditos mutuamente visíveis se o segmento de reta que os conecta não possui outros pontos azuis. Prove que existe um conjunto de 2019 pontos azuis que são mutuamente visíveis dois a dois.[/hide] PS. There is a comment about problem being wrong / incorrect [url=https://artofproblemsolving.com/community/c6h1957974p14780265]here[/url]

Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?

5)Players $A$ and $B$ play the following game: a player writes, in a board, a positive integer $n$, after this they delete a number in the board and write a new number where can be: i)The last number $p$, where the new number will be $p - 2^k$ where $k$ is greatest number such that $p\ge 2^k$ ii) The last number $p$, where the new number will be $\frac{p}{2}$ if $p$ is even. The players play alternately, a player win(s) if the new number is equal to $0$ and player $A$ starts. a)Which player has the winning strategy with $n = 40$?? b)Which player has the winning strategy with $n = 2012$??

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

Determine the greatest positive integer $A{}$ with the following property: however we place the numbers $1,2,\ldots, 100$ on a $10\times 10$ board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least $A{}$.

For odd integers $n,$ two people play the game on $2\times n$ grid. Each people color one cell that is not colored before with white and black. When coloring is done, they count the number of ordered pairs of neighboring cells that have the same color and different color, respectively. If same color neighboring ordered pair of cells are more than different color neighboring ordered pair of cells, the person who first starts win and lose otherwise. (If the number is same, they are tied.) If both of them use the best strategy, find the result of the game.

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? $\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$

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.)

Let N be a positive integer greater than 2. We number the vertices of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N + 1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way. In the first step we mark the vertex 1. If ni is the vertex marked in the i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away from vertex ni, counting clockwise if ni is positive and counter-clockwise if ni is negative. This procedure is repeated till we reach a vertex that has already been marked. Let $f(N)$ be the number of non-marked vertices. (a) If $f(N) = 0$, prove that 2N + 1 is a prime number. (b) Compute $f(1997)$.

Paraflea makes jumps on the plane, starting from the origin $(0, 0)$. From point $(x, y)$ it may jump to another point of the form $(x + p, y + p^2)$, where $p$ is any positive real number. (The value of $p$ may differ for each jump.) a) Is there any point in quadrant $I$ which cannot be reached by the flea? (Quadrant $I$ contains points $(x, y)$ for which $x$ and $y$ are positive real numbers.) b) What is the minimum number of jumps that the flea must make from the origin so that it gets to the point $(100, 1)$?

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

Given an isosceles trapezoid and draw one of its diagonals (so we get $2$ triangles). On each triangle we get, there is an ant walking along the edge. There speed are equal and unchanged. These $2$ ants also didn't change their walking directions and their directions are opposite. Prove that: for all initial locations of the ants, they will meet each other at some time.

Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.

Jenny is going to attend a sports camp for $7$ days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of $4$ consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.

Supppose that there are roads $AB$ and $CD$ but there are no roads $BC$ and $AD$ between four cities $A$, $B$, $C$, and $D$. Define [i]restructing[/i] to be the changing a pair of roads $AB$ and $CD$ to the pair of roads $BC$ and $AD$. Initially there were some cities in a country, some of which were connected by roads and for every city there were exactly $100$ roads starting in it. The minister drew a new scheme of roads, where for every city there were also exactly $100$ roads starting in it. It's known also that in both schemes there were no cities connected by more than one road. Prove that it's possible to obtain the new scheme from the initial after making a finite number of restructings. [i] (Т. Зубов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$. Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!

Given an integer $n\geq 2$. Call a positive integer ${T}$ [i]Pingsheng Number[/i], if there exists pairwise different non empty subsets $A_1,A_2,\cdots ,A_m$ $(m\geq 3)$ of set $S=\{1,2,\cdots ,n\},$ satisfying $T=\sum\limits_{i=1}^m|A_i|,$ and for $\forall p,q,r\in\{1,2,\cdots ,m\},p\neq q,q\neq r,r\neq p,$ we have $A_p\cap(A_q\triangle A_r)=\varnothing$ or $A_p\subseteq (A_q\triangle A_r).$ Find the max [i]Pingsheng Number[/i].