In a football tournament with $n\geq 2$ teams each two played a match. For a won match the victor gets 2 points and for a draw each one gets 1 point. In the final results there weren’t two teams with equal amount of points. It turned out that because of a mistake each match that was written in the results as won was actually a draw and each one that was written as draw was actually won. In the new ranking there were also no two teams with the same amount of points. Find all n for which it is possible for the two rankings to be opposite of each other, that is the first team in the first ranking is actually the last one, the second team is pre-last and so on.

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. If $e(G_{n}) \geq\frac{n\sqrt{n}}{2}+\frac{n}{4}$ ,prove that $G_{n}$ contains $C_{4}$ .

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

Let $n$ and $k$ be positive integers. The square in the $i$th row and $j$th column of an $n$-by-$n$ grid contains the number $i+j-k$. For which $n$ and $k$ is it possible to select $n$ squares from the grid, no two in the same row or column, such that the numbers contained in the selected squares are exactly $1,\,2,\,\ldots,\,n$?

This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url]. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

There are $30$ boxes each with a unique key. The keys are randomly arranged in the boxes, so that each box contains just one key and the boxes are locked. Two boxes are broken open, thus releasing two keys. What is the probability that the remaining boxes can be opened without forcing them?

Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].

There are $1000$ inhabitants in a settlement. Every evening every inhabitant tells all his friends all the news he had heard the previous day. Every news becomes finally known to every inhabitant. Prove that it is possible to choose $90$ of inhabitants so, that if you tell them a news simultaneously, it will be known to everybody in $10$ days.

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$. [b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$? [b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$? [b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction? [b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$? [b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$? [b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$ [b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$. [b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards? [b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)? [b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.) [b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees. [b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$? [b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$? [b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat? [b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.) [b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$. [b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying $$\begin{cases} -x^2 + 3y^2 - 5x + 7y + 4 = 0 \\ 2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$ [b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish? [b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.

Let $P$ be a cyclic polygon with circumcenter $O$ that does not lie on any diagonal, and let $S$ be the set of points on 2D plane containing $P$ and $O$. The $\textit{Matcha Sweep Game}$ is a game between two players $A$ and $B$, with $A$ going first, such that each choosing a nonempty subset $T$ of points in $S$ that has not been previously chosen, and such that if $T$ has at least $3$ vertices then $T$ forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. For which polygons $P$ can $A$ guarantee a win? [i]Proposed by Anzo Teh Zhao Yang[/i]

Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.

[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer. [b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.) [b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality: (a) $xy \le \left(\frac{x+y}{2}\right)^2$ (b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$ [b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img] [b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We have 101 coins and a two-pan scale. In one weighing, we can compare the weights of two coins. What is the smallest number of weighings required in order to decide whether there exist 51 coins which all have the same weight?

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A group of persons is called [i]good[/i] if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other. A group is called [i]perfect[/i] if it is good and every set of its members is also good. A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect. [i]Author: C. Berge[/i]

Find all possible values of $n$ for which a rectangular board $9\times n$ can be partitioned into tiles of the shape: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8wLzdjM2Y4ZmE0Zjg1YWZlZGEzNTQ1MmEyNTc3ZjJkNzBlMjExYmY1LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yMiBhdCA1LjEzLjU3IEFNLnBuZw==[/img]

There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

Let $G$ be a complete directed graph with $100$ vertices such that for any two vertices $x,y$ one can find a directed path from $x$ to $y$. a) Show that for any such $G$, one can find a $m$ such that for any two vertices $x,y$ one can find a directed path of length $m$ from $x$ to $y$ (Vertices can be repeated in the path) b) For any graph $G$ with the properties above, define $m(G)$ to be smallest possible $m$ as defined in part a). Find the minimim value of $m(G)$ over all such possible $G$'s.

A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$ $(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$ $(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$

In a summer camp there are $n$ girls $D_1,D_2, ... ,D_n$ and $2n-1$ boys $C_1,C_2, ...,C_{2n-1}$. The girl $D_i, i = 1,2,... ,n,$ knows only the boys $C_1,C_2, ... ,C_{2i-1}$. Let $A(n, r)$ be the number of different ways in which $r$ girls can dance with $r$ boys forming $r$ pairs, each girl with a boy she knows. Prove that $A(n, r) = \binom{n}{r} \frac{r!}{(n-r)!}.$

Given two positive integers $n$ and $m$ and a function $f : \mathbb{Z} \times \mathbb{Z} \to \left\{0,1\right\}$ with the property that \begin{align*} f\left(i, j\right) = f\left(i+n, j\right) = f\left(i, j+m\right) \qquad \text{for all } \left(i, j\right) \in \mathbb{Z} \times \mathbb{Z} . \end{align*} Let $\left[k\right] = \left\{1,2,\ldots,k\right\}$ for each positive integer $k$. Let $a$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i+1, j\right) = f\left(i, j+1\right) . \end{align*} Let $b$ be the number of all $\left(i, j\right) \in \left[n\right] \times \left[m\right]$ satisfying \begin{align*} f\left(i, j\right) = f\left(i-1, j\right) = f\left(i, j-1\right) . \end{align*} Prove that $a = b$.

Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions: $\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$. $\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$. $\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$. Note: One or more of the sets may be empty.

Is it possible to inscribe an $N$-gon in a circle so that all the lengths of its sides are different and all its angles (in degrees) are integer, where a) $N = 19$, b) $N = 20$ ? Mikhail Malkin