Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property? (A Shapovalov)

Positive integer numbers $n$ and $k > 1$ are given. Losyash likes some of the cells of the $n \times n$ checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of $2n + 2$, the upper-left corner of which coincides with the upper-left corner of the board (there are $n$ such rectangles in total). Given $n$ and $k$, determine whether Losyash can color each cell he likes in one of $k$ colors so that in any rectangle of interest to him the number of cells of any two colors differ by no more than $1$.

$N$ men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all $N$.

$1956$ points are chosen in a cube with edge $13$. Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?

For all positive integers $m\geq 1$, denote by $\mathcal{G}_m$ the set of simple graphs with exactly $m$ edges. Find the number of pairs of integers $(m,n)$ with $1<2n\leq m\leq 100$ such that there exists a simple graph $G\in\mathcal{G}_m$ satisfying the following property: it is possible to label the edges of $G$ with labels $E_1$, $E_2$, $\ldots$, $E_m$ such that for all $i\neq j$, edges $E_i$ and $E_j$ are incident if and only if either $|i-j|\leq n$ or $|i-j|\geq m-n$. $\textit{Note: }$ A graph is said to be $\textit{simple}$ if it has no self-loops or multiple edges. In other words, no edge connects a vertex to itself, and the number of edges connecting two distinct vertices is either $0$ or $1$.

A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five. Using the four words “Hi”, “hey”, “hello”, and “haiku”, How many haikus Can somebody make? (Repetition is allowed, Order does matter.) [i]Proposed by Jeff Lin[/i]

Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?

Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.

Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are $20$ cookies on a red plate and $14$ on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.

Let $N$ be the number of sequences of natural numbers $d_1,d_2,\dots,d_{10}$ such that the following conditions hold: $d_1|d_2$, $\dots$, $d_9|d_{10}$ and $d_{10}|6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.

For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that $C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.

A box contains red, blue, and white balls, $100$ balls in total. It is known that among any $26$ of them there are always $10$ balls of the same color. Find the minimal number $N$ such that among any $N$ balls there are always $30$ balls of the same color.

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

Suppose that $n - 1$ items $A_1,A_2,...,A_{n-1}$ have already been arranged in the increasing order, and that another item $A_n$ is to be inserted to preserve the order. What is the expected number of comparisons necessary to insert $A_n$?

Given a piece of graph paper with a letter assigned to each vertex of every square such that on every segment connecting two vertices that have the same letter and are on the same line of the mesh, there is at least one vertex with another letter. What is the least number of distinct letters needed to plot such a picture, along the sides of the cells?

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_0, P_1, \ldots, P_N$ in the plane. After choosing $P_k$ (but before choosing $P_{k+1}$) for $k \geq 1$, Bob tells "warmer'' if $P_k$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder''. After Alice has chosen $P_N$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $AB$ is at most $\frac 1 {2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

For a positive integer $n$ define a sequence of zeros and ones to be [i]balanced[/i] if it contains $n$ zeros and $n$ ones. Two balanced sequences $a$ and $b$ are [i]neighbors[/i] if you can move one of the $2n$ symbols of $a$ to another position to form $b$. For instance, when $n = 4$, the balanced sequences $01101001$ and $00110101$ are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set $S$ of at most $\frac{1}{n+1} \binom{2n}{n}$ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in $S$.

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

[u]Round 1[/u] [b]p1.[/b] Ten children arrive at a birthday party and leave their shoes by the door. All the children have different shoe sizes. Later, as they leave one at a time, each child randomly grabs a pair of shoes their size or larger. After some kids have left, all of the remaining shoes are too small for any of the remaining children. What is the greatest number of shoes that might remain by the door? [b]p2.[/b] Turans, the king of Saturn, invented a new language for his people. The alphabet has only $6$ letters: A, N, R, S, T, U; however, the alphabetic order is different than in English. A word is any sequence of $6$ different letters. In the dictionary for this language, the first word is SATURN. Which word follows immediately after TURANS? [b]p3.[/b] Benji chooses five integers. For each pair of these numbers, he writes down the pair's sum. Can all ten sums end with different digits? [b]p4.[/b] Nine dwarves live in a house with nine rooms arranged in a $3\times3$ square. On Monday morning, each dwarf rubs noses with the dwarves in the adjacent rooms that share a wall. On Monday night, all the dwarves switch rooms. On Tuesday morning, they again rub noses with their adjacent neighbors. On Tuesday night, they move again. On Wednesday morning, they rub noses for the last time. Show that there are still two dwarves who haven't rubbed noses with one another. [b]p5.[/b] Anna and Bobby take turns placing rooks in any empty square of a pyramid-shaped board with $100$ rows and $200$ columns. If a player places a rook in a square that can be attacked by a previously placed rook, he or she loses. Anna goes first. Can Bobby win no matter how well Anna plays? [img]https://cdn.artofproblemsolving.com/attachments/7/5/b253b655b6740b1e1310037da07a0df4dc9914.png[/img] [u]Round 2[/u] [b]p6.[/b] Some boys and girls, all of different ages, had a snowball fight. Each girl threw one snowball at every kid who was older than her. Each boy threw one snowball at every kid who was younger than him. Three friends were hit by the same number of snowballs, and everyone else took fewer hits than they did. Prove that at least one of the three is a girl. [b]p7.[/b] Last year, jugglers from around the world travelled to Jakarta to participate in the Jubilant Juggling Jamboree. The festival lasted $32$ days, with six solo performances scheduled each day. The organizers noticed that for any two days, there was exactly one juggler scheduled to perform on both days. No juggler performed more than once on a single day. Prove there was a juggler who performed every day. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?

In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of $ T $ such that for each point $ (x;y)\in T $ then either $ (x;y) $ or $ (-x;-y) $ is coloured. Denote $ N $ to be the number of couples $ {(x_1;y_1),(x_2;y_2)} $ such that both $ (x_1;y_1) $ and $ (x_2;y_2) $ are coloured and $ x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41} $. Find the all possible values of $ N $.

Determine all pairs of positive integers $(m,n)$ for which an $m\times n$ rectangle can be tiled with (possibly rotated) L-shaped trominos.

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)