A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$. Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows: A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.

Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$

A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning. Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).

A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.

Given a positive integer $n$. Consider a $3 \times n$ grid, a set $S$ of squares is called [i]connected[/i] if for any points $A \neq B$ in $S$, there exists an integer $l \ge 2$ and $l$ squares $A=C_1,C_2,\dots ,C_l=B$ in $S$ such that $C_i$ and $C_{i+1}$ shares a common side ($i=1,2,\dots,l-1$). Find the largest integer $K$ satisfying that however the squares are colored black or white, there always exists a [i]connected[/i] set $S$ for which the absolute value of the difference between the number of black and white squares is at least $K$.

Two magicians are performing the following game. Initially the first magician encloses the second magician in a cabin where he can neither see nor hear anything. To start the game, the first magician invites Daniel, from the audience, to put on each square of a chessboard $n \times n$, at his (Daniel's) discretion, a token black or white. Then the first magician asks Daniel to show him a square $C$ of his own choice. At this point, the first magician chooses a square $D$ (not necessarily different from $C$) and replaces the token that is on $D$ with other color token (white with black or black with white). Then he opens the cabin in which the second magician was held. Looking at the chessboard, the second magician guesses what is the square $C$. For what value of $n$, the two magicians have a strategy such that the second magician makes a successful guess.

Let $m> 3$ be an integer. At a camp there are more than $m$ participants. The camp manager discovers that every time he picks out the camp participants, they say they have exactly one mutual friend among the participants. Which is the largest possible number of participants at the camp? (If $A$ is a friend of $B, B$ is also a friend of $A$. A person is not considered a friend of himself.)

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two people throw coins. One throws his coin $10$ times, the other throws his $11$ times . What is the probability that the second coin fell showing "heads" a greater number of times than the first? (For those not familiar with Probability Theory this question could have been reformulated thus : Consider various arrangements of a $21$ digit number in which each digit must be a " $1$ " or a "$2$" . Among all such numbers what fraction of them will have more occurrences of the digit "$2$" among the last $11$ digits than among the first $10$?) (S. Fomin , Leningrad)

A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$. Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.

There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.

A [i]fortress[/i] is a finite collection of cells in an infinite square grid with the property that one can pass from any cell of the fortress to any other by a sequence of moves to a cell with a common boundary line (but it can have "holes"). The [i]walls[/i] of a fortress are the unit segments between cells belonging to the fortress and cells not belonging to the fortress. The [i]area[/i] $A$ of a fortress is the number of cells it consists of. The [i]perimeter[/i] $P$ is the total length of its walls. Each cell of the fortress can contain a [i]guard[/i] which can oversee the cells to the top, the bottom, the right and the left of this cell, up until the next wall (it also oversees its own cell). (a) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of perimeter $P \le 2024$. (b) Determine the smallest integer $k$ such that $k$ guards suffice to oversee all cells of any fortress of area $A \le 2024$.

Let $n$ be a positive integer. Ana and Banana are playing the following game: First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana, where each command consists of swapping two adjacent cups in the row. Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information about the position of the hole and the position of the ball at any point, what is the smallest number of commands she has to give in order to achieve her goal?

An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of \[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\] How many guesses are needed for Banana to figure out the quadruple Ana chose?

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having $n$ inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability $1/2$ independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let $p_n$ denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that $p_n\to 1$ if $n\to\infty$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Let be four positive integers $m, n, p, q$, with $p < m$ given and $q < n$. Take four points $A(0; 0), B(p; 0), C (m; q)$ and $D(m; n)$ in the coordinate plane. Consider the paths $f$ from $A$ to $D$ and the paths $g$ from $B$ to $C$ such that when going along $f$ or $g$, one goes only in the positive directions of coordinates and one can only change directions (from the positive direction of one axe coordinate into the the positive direction of the other axe coordinate) at the points with integral coordinates. Let $S$ be the number of couples $(f, g)$ such that $f$ and $g$ have no common points. Prove that \[S = \binom{n}{m+n} \cdot \binom{q}{m+q-p} - \binom{q}{m+q} \cdot \binom{n}{m+n-p}.\]

In a circular circuit there are petrol stations, so that the total accumulated petrol in them it is exactly enough for a car to go around the circuit. Prove that there is a position from where a car, with the tank of finite capacity and initially empty, can leave and get to go a full loop around the circuit, stopping to refuel at positions. [hide=original wording]En un circuito circular hay puestos de gasolina, de modo que el total de la gasolina acumulada en ellos es exactamente suciente para que un auto de una vuelta completa al circuito. Demostrar que existe un puesto desde donde un auto, con el estanque de capacidad finita e inicialmente vacio, puede partir y conseguir recorrer una vuelta completa al circuito, deteniendose a reabastecerse de gasolina en los puestos.[/hide]

In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules:\\ $1$. It must be in the interior of the island (none on borders).\\ $2$. No two airports can be at the exact same location.\\ $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle.\\ $4$. No three airports can be collinear.\\ Find the maximum value of $k$ for each $n$ [hide=Harder Version]Replace $1$ new and $2$ old with $1$ old and $2$ new.[/hide]

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.

Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$.