Faces of a cube $9\times 9\times 9$ are partitioned onto unit squares. The surface of a cube is pasted over by $243$ strips $2\times 1$ without overlapping. Prove that the number of bent strips is odd. [i]A. Poliansky[/i]

Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.

A tournament has $5$ players and is in round-robin format (each player plays each other exactly once). Each game has a $\frac13$ chance of player $A$ winning, a $\frac13$ chance of player $B$ winning, and a$ \frac13$ chance of ending in a draw. The probability that at least one player draws all of their games can be written in simplest form as $\frac{m}{3^n}$ where $m, n$ are positive integers. Find $m + n$.

Arnulfo and Berenice play the following game: One of the two starts by writing a number from $ 1$ to $30$, the other chooses a number from $ 1$ to $30$ and adds it to the initial number, the first player chooses a number from $ 1$ to $30$ and adds it to the previous result, they continue doing the same until someone manages to add $2018$. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question: Adding chosen numbers from $1 $ to $a$, until reaching the number $ b$, what conditions must meet $a$ and $ b$ so that the first player does not have a winning strategy? Indicate if Arnulfo and Berenice are right and answer the question asked by them.

For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.

In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes. [i]Proposed by Mohammad Moshtaghi Far, Kian Shamsaie[/i] [b]Rated 4[/b]

Ana has a scale that shows which side weight more or if both side are equal. She has $4$ weights which look the same but they weight $1001g, 1002g, 1004g$ and $1005g$, respectively. Is it possible for Ana to find out the weight of each of them with only $4$ measurements?

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$

Prove the identities: $$C_{n}^{1}+2C_{n}^{2}+3C_{n}^{3}+...+nC_{n}^{n}=n\cdot 2 ^{n-1}$$ $$C_{n}^{1}-2C_{n}^{2}+3C_{n}^{3}+...-(-1)^{n-1}nC_{n}^{n}=0$$

Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.

Lucky starts doodling on a $5\times 5$ Bingo board. He puts his pencil at the center of the upper-left square (marked by ‘·’) and draws a continuous doodle ending on the Free Space, never going off the board or through a corner of a square. (See Figure 1.) (a) Is it possible for Lucky’s doodle to visit all squares exactly once? (The starting and ending squares are considered visited.) (b) Is it possible for Lucky’s doodle to visit all squares exactly twice?

There is number $1$ on the blackboard initially. The first step is to erase $1$ and write two nonnegative reals whose sum is $1$. Call the smaller number of the two $L_2$. For integer $k \ge 2$, the ${k}$ the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the $k+1$ on the blackboard $L_{k+1}$. Find the maximum of $L_2+L_3+\cdots+L_{2024}$.

Bessie the cow is hungry and wants to eat some oranges, which she has an infinite supply of. Bessie starts with a fullness level of $0$, and each orange that she eats increases her fullness level by $85$. She can also eat lemons, and each time she eats a lemon, her fullness level is halved, rounding down. What is the smallest number of lemons that Bessie should have in order to be able to attain every possible nonnegative integer fullness level?

Given positive integer $m,n$, color the points of the regular $(2m+2n)$-gon in black and white, $2m$ in black and $2n$ in white. The [i]coloring distance[/i] $d(B,C) $ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points. The [i]coloring distance[/i] $d(W,X) $ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points. We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon. We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon. We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j) $. Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Let $ G$ be a simple graph with vertex set $ V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\}$ .$ j$and$ j\plus{}1$ are connected by an edge for $ 0\le j\le n$. Let $ A$ be a subset of $ V$ and $ G(A)$ be the induced subgraph associated with $ A$. Let $ O(G(A))$ be number of components of $ G(A)$ having an odd number of vertices. Let $ T(p,r)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\}$ for $ r\le p \le 2r$. Prove That $ |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}$.

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

$5$ teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains $5$ points and a losing team $0$ points. For a $0-0$ draw both teams gain $1$ point, and for other draws ($1-1,2-2,3-3,$etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?

The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let $M$ be an arbitrary set of $n$ squares of this net. Prove that it is possible to choose no fewer than $n/4$ squares of $M$ in such a way that no two of them have a common point.

A lake is in the shape of a regular hexagon of side length \(1\). Initially there is a single lotus leaf somewhere in the lake, sufficiently far from the shore. Each day, from every existing leaf a new leaf may grow at distance \(\sqrt{3}\) (measured between centers), provided it does not overlap any other leaf. If the lake is large enough that edge effects never interfere, what is the least number of days required to have \(2025\) leaves in the lake?

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?

Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? [i]Proposed by Ivan Chan Kai Chin[/i]

Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. When at the beginning of the turn there are $n\geq 1$ marbles on the table, then the player whose turn it is removes $k$ marbles, where $k\geq 1$ either is an even number with $k\leq \frac{n}{2}$ or an odd number with $\frac{n}{2}\leq k\leq n$. A player win the game if she removes the last marble from the table. Determine the smallest number $N\geq 100000$ such that Berta can enforce a victory if there are exactly $N$ marbles on the tale in the beginning.