A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]

A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.

[b]p1.[/b] (See the diagram below.) $ABCD$ is a square. Points $G$, $H$, $I$, and $J$ are chosen in the interior of $ABCD$ so that: (i) $H$ is on $\overline{AG}$, $I$ is on $\overline{BH}$, $J$ is on $\overline{CI}$, and $G$ is on $\overline{DJ}$ (ii) $\vartriangle ABH \sim \vartriangle BCI \sim \vartriangle CDJ \sim \vartriangle DAG$ and (iii) the radii of the inscribed circles of $\vartriangle ABH$, $\vartriangle BCI$, $\vartriangle CDJ$, $\vartriangle DAK$, and $GHIJ$ are all the same. What is the ratio of $\overline{AB}$ to $\overline{GH}$? [img]https://cdn.artofproblemsolving.com/attachments/f/b/47e8b9c1288874bc48462605ecd06ddf0f251d.png[/img] [b]p2.[/b] The three solutions $r_1$, $r_2$, and $r_3$ of the equation $$x^3 + x^2 - 2x - 1 = 0$$ can be written in the form $2 \cos (k_1 \pi)$, $2 \cos (k_2 \pi)$, and $2 \cos (k_3 \pi)$ where $0 \le k_1 < k_2 < k_3 \le 1$. What is the ordered triple $(k_1, k_2, k_3)$? [b]p3.[/b] $P$ is a convex polyhedron, all of whose faces are either triangles or decagons ($10$-sided polygon), though not necessarily regular. Furthermore, at each vertex of $P$ exactly three faces meet. If $P$ has $20$ triangular faces, how many decagonal faces does P have? [b]p4.[/b] $P_1$ is a parabola whose line of symmetry is parallel to the $x$-axis, has $(0, 1)$ as its vertex, and passes through $(2, 2)$. $P_2$ is a parabola whose line of symmetry is parallel to the $y$-axis, has $(1, 0)$ as its vertex, and passes through $(2, 2)$. Find all four points of intersection between $P_1$ and $P_2$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.

Find the number of functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ having the property that $ (f\circ f\circ f)(n)=n+3, $ for any natural numbers $ n. $

Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step. d) Whoever reaches the last step (listed with $2022$) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.

How many six-digit perfect squares can be formed using all the numbers $1,2,3,4,5,6$ as digits? $\textbf{(A)}~5$ $\textbf{(B)}~19$ $\textbf{(C)}~7$ $\textbf{(D)}~\text{None of the above}$

In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even

Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].

Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled? $\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$

A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$, $B$, and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$, $p_2$, $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Prove that \[ \lim_{N \to \infty} \frac{a(N)}{N} \] exists and determine its value.

Determine whether there exists an arithimethical progression consisting of 40 terms and each of whose terms can be written in the form $ 2^m \plus{} 3^n$ or not. where $ m,n$ are nonnegative integers.

The natural numbers $0, 1, 2, 3, . . .$ are written on the square table $2015\times 2015$ in a circular order (anti-clockwise) such that $0$ is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number $2015$ is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number $0$ in the center by $14$, after that, each time, we can add $1$ to each of $12$ numbers on $12$ consecutive unit squares in a row, or $12$ consecutive unit squares in a column, or $12$ unit squares in a rectangle $3\times 4$. After a finite number of steps, can we make all numbers on the table are multiples of $2016$? [img]https://cdn.artofproblemsolving.com/attachments/c/d/223b32c0e3f58f62d0d40fa78c09a2cd035ed5.png[/img]

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$? [i]Proposed by Morteza Saghafian[/i]

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

In each lattice-point of an $m \times n$ grid and in the centre of each of the formed unit squares a pawn is placed. a) Find all such grids with exactly $500$ pawns. b) Prove that there are infinitely many positive integers $k$ for which therer is no grid containing exactly $k$ pawns.

In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-\left(2^5-2^6-2^7\right)\right)- \left(2^8- 2^9 \right)-2^{10}= -933$ How many different results can we obtain?

In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that: [LIST] [*] There are no two numbers written in the first row that are equal to each other.[/*] [*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*] [*] The two numbers written in each column are different and they add up to a rational number.[/*] [/LIST] Determine the maximum quantity of irrational numbers that can be in the chessboard.

It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$

A solid figure consisting of unit cubes is shown in the picture. Is it possible to exactly fill a cube with these figures if the side length of the cube is a) 15; b) 30?

Given a family $C$ of circles of the same radius $R$, which completely covers the plane (that is, every point in the plane belongs to at least one circle of the family), prove that there exist two circles of the family such that the distance between their centers is less than or equal to $R\sqrt3$ .