In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. [img]https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png[/img]

A convex polygon of sides $\ell_1, \ell_2, ..., \ell_n$ is called [i]ordered [/i] if for all reordering $( \sigma (1), \sigma (2), ..., \sigma (n))$ of the set $(1, 2,..., n)$ there exists a point $P$ inside the polygon such that $d_{\sigma (1)} < _{\sigma (2)} <...< d_{\sigma (n)}$ , where $d_i$ represents the distance between $P$ and side $\ell_i$. Find all the convex ordered polygons.

Let $n \geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1, 2,\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number $1$. Into each of the triangles the product of the numbers on its sides is written. Let S be the sum of those $n - 2$ products. Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.

Is it possible to write a positive integer in every cell of an infinite chessboard, in such a manner that, for all positive integers $m, n$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n$ ?

A convex polygon in the Cartesian plane has all of its vertices on integer coordinates. One of the sides of the polygon is $AB$ where $A = (0, 0)$ and $B = (51, 51)$, and the interior angles at $A$ and $B$ are both at most $45$ degrees. Assuming no $180$ degree angles, what is the maximum number of vertices this polygon can have?

Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?

Can we find $N$ such that all $m\times n$ rectangles with $m,n>N$ can be tiled with $4\times6$ and $5\times7$ rectangles?

A rectangular grid whose side lengths are integers greater than $1$ is given. Smaller rectangles with area equal to an odd integer and length of each side equal to an integer greater than $1$ are cut out one by one. Finally one single unit is left. Find the least possible area of the initial grid before the cuttings. Ps. Collected [url=https://artofproblemsolving.com/community/c949611_2019_nigerian_senior_mo_round_3]here[/url]

Gandalf (the wizard) and Bilbo (the assistant) are presenting a magic trick to Nitzan (the audience). While Gandalf leaves the room, Nitzan chooses a number $1\leq x\leq 2^{2022}$ and shows it to Bilbo. Now bilbo writes on the board a long row of $N$ digits, each of which is $0$ or $1$. After this Nitzan can, if he wishes, switch the order of two consecutive digits in the row, but only once. Then Gandalf returns to the room, looks at the row, and guesses the number $x$. Can Bilbo and Gandalf come up with a strategy that allows Gandalf to guess $x$ correctly no matter how Nitzan acts, if [b]a)[/b] $N=2500$? [b]b)[/b] $N=2030$? [b]c)[/b] $N=2040$?

A regular triangle has side length $n{}$. We divided its sides into $n{}$ equal parts and drew a line segment parallel with each side through the dividing points. A lattice of $1+2+\ldots+(n+1)$ intersection points is thus formed. For which positive integers $n{}$ can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length $1$? [i]Proposed by Alexander Gunning, Cambridge, UK[/i]

An [i]$n$-type triangle[/i] where $n\geqslant 2$ is formed by the cells of a $(2n+1)\times(2n+1)$ board, situated under both main diagonals. For instance, a $3$-type triangle looks like this:[img]https://i.ibb.co/k4fmwWY/Screenshot-2024-07-31-153932.png[/img]Determine the maximal length of a sequence with pairwise distinct cells in an $n$-type triangle, such that, beggining with the second one, any cell of the sequence has a common side with the previous one. [i]Cristi Săvescu[/i]

Given a table in a form of the regular $1000$-gon with sidelength $1$. A Beetle initially is in one of its vertices. All $1000$ vertices are numbered in some order by numbers $1$, $2$, $\ldots$, $1000$ so that initially the Beetle is in the vertex $1$. The Beetle can move only along the edges of $1000$-gon and only clockwise. He starts to move from vertex $1$ and he is moving without stops until he reaches vertex $2$ where he has a stop. Then he continues his journey clockwise from vertex $2$ until he reaches the vertex $3$ where he has a stop, and so on. The Beetle finishes his journey at vertex $1000$. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to $2017$.

The boba shop sells four different types of milk tea, and William likes to get tea each weekday. If William refuses to have the same type of tea on successive days, how many different combinations could he get, Monday through Friday?

Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).) Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee? [i]Proposed by Espen Slettnes[/i]

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$ how many such colorations are possible?

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

$n$ people live in a town, and they are members of some clubs (residents can be members of more than one club). No matter how we choose some (but at least one) clubs, there is a resident of the town who is the member of an odd number of the chosen clubs. Prove that the number of clubs is at most $n$. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine. Prove that, on one of these $16$ days, all seven dwarves were collecting berries.

A polygonal line of the length $1001$ is given in a unit square. Prove that there exists a line parallel to one of the sides of the square that meets the polygonal line in at least $500$ points.

Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An [i]operation[/i] is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations. [i]Proposed by Zhenhua Qu[/i]

An infinite fairytale is a book with pages numbered $1,2,3,\ldots$ where all natural numbers appear. An author wants to write an infinite fairytale such that a new dwarf is introduced on each page. Afterward, the page contains several discussions between groups of at least two of the already introduced dwarfs. The publisher wants to make the book more exciting and thus requests the following condition: Every infinite set of dwarfs contains a group of at least two dwarfs, who formed a discussion group at some point as well as a group of the same size for which this is not true. Can the author fulfill this condition?

Ivan writes the matrix $\begin{pmatrix} 2 & 3\\ 2 & 4\end{pmatrix}$ on the board. Then he performs the following operation on the matrix several times: [b]1.[/b] he chooses a row or column of the matrix, and [b]2.[/b] he multiplies or divides the chosen row or column entry-wise by the other row or column, respectively. Can Ivan end up with the matrix $\begin{pmatrix} 2 & 4\\ 2 & 3\end{pmatrix}$ after finitely many steps?

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy: i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\ ii) $ f(g(x))=f(x) , \forall x \in S $\\ iii) $f(x+n)=f(x) ,\forall x \in S$\\ iv) $ g $ is bijective.\\ Find the number of pair of such functions $ (f,g)$ for every $n$.