In a sports tournament involving $N$ teams, each team plays every other team exactly one. At the end of every match, the winning team gets $1$ point and losing team gets $0$ points. At the end of the tournament, the total points received by the individual teams are arranged in decreasing order as follows: \[x_1 \ge x_2 \ge \cdots \ge x_N . \] Prove that for any $1\le k \le N$, \[\frac{N - k}{2} \le x_k \le N - \frac{k+1}{2}\]

$n$ numbers are written down along the circumference. Their sum equals to zero, and one of them equals $1$. a) Prove that there are two neighbours with their difference not less than $n/4$. b) Prove that there is a number that differs from the arithmetic mean of its two neighbours not less than on $8/(n^2)$. c) Try to improve the previous estimation, i.e what number can be used instead of $8$? d) Prove that for $n=30$ there is a number that differs from the arithmetic mean of its two neighbours not less than on $2/113$, give an example of such $30$ numbers along the circumference, that not a single number differs from the arithmetic mean of its two neighbours more than on $2/113$.

A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition. Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles. [i]Proposed by S. Volchenkov[/i]

There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of $n$.

For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$, \[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\] [i]Alex Zhu.[/i]

Let $n$ be a positive integer. A square $ABCD$ is divided into $n^2$ identical small squares by drawing $(n-1)$ equally spaced lines parallel to the side $AB$ and another $(n- 1)$ equally spaced lines parallel to $BC$, thus giving rise to $(n+1)^2$ intersection points. The points $A, C$ are coloured red and the points $B, D$ are coloured blue. The rest of the intersection points are coloured either red or blue. Prove that the number of small squares having exactly $3$ vertices of the same colour is even.

Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles

Ten people apply for a job. The manager decides to interview the candidates one by one according to the following conditions: i) the first three candidates will not be employed; ii) from the fourth candidates onwards, if a candidate's comptence surpasses the competence of all those who preceded him, then that candidate is employed; iii) if the first nine candidates are not employed, then the tenth candidate will be employed. We assume that none of the $10$ applicants have the same competence, and these competences can be ranked from the first to tenth. Let $P_k$ represent the probability that the $k$th-ranked applicant in competence is employed. Prove that: i) $P_1>P_2>\ldots>P_8=P_9=P_{10}$; ii) $P_1+P_2+P_3>0.7$ iii) $P_8+P_9+P_{10}\le 0.1$. [i]Su Chun[/i]

Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas. What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?

It is considered a regular polygon with $n$ sides, where $n(n>3)$ is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen $m(0\leq m\leq n)$ vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered $monocolor$ ,if all of its vertices are of the same color. Prove that the number of all $monocolor$ isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these $monocolor$ isosceles triangles.

Let $k$ be a positive integer and $N_k$ be the number of sequences of length $2001$, all members of which are elements of the set $\{0,1,2,\ldots,2k+1\}$, and the number of zeroes among these is odd. Find the greatest power of $2$ which divides $N_k$.

Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$. [i]David Yang.[/i]

The numbers $1, 2, 3, . . . , 101$ are written in a row in some order. Prove that it is always possible to erase $90 $ of the numbers so that the remaining $11$ numbers remain arranged in either increasing or decreasing order.

For a natural number $ n, $ a string $ s $ of $ n $ binary digits and a natural number $ k\le n, $ define an $ n,s,k$ [i]-block[/i] as a string of $ k $ consecutive elements from $ s. $ We say that two $ n,s,k\text{-blocks} , $ namely, $ a_1a_2\ldots a_k,b_1b_2\ldots b_k, $ are [i]incompatible[/i] if there exists an $ i\in\{1,2,\ldots ,k\} $ such that $ a_i\neq b_i. $ Also, for two natural numbers $ r\le n, l, $ we say that $ s $ is $ r,l $ [i]-typed[/i] if there are, at most, $ l $ pairwise incompatible $ n,s,r\text{-blocks} . $ Let be a $ 3,7\text{-typed} $ string $ t $ consisting of $ 10000 $ binary digits. Determine the maximum number $ M $ that satisfies the condition that $ t $ is $ 10,M\text{-typed} . $ [i]Cătălin Gherghe[/i]

[u]Round 1[/u] [b]p1.[/b] Evaluate the sum $1-2+3-...-208+209-210$. [b]p2.[/b] Tony has $14$ beige socks, $15$ blue socks, $6$ brown socks, $8$ blond socks and $7$ black socks. If Tony picks socks out randomly, how many socks does he have to pick in order to guarantee a pair of blue socks? [b]p3.[/b] The price of an item is increased by $25\%$, followed by an additional increase of $20\%$. What is the overall percentage increase? [u]Round 2[/u] [b]p4.[/b] A lamp post is $20$ feet high. How many feet away from the base of the post should a person who is $5$ feet tall stand in order to cast an 8-foot shadow? [b]p5.[/b] How many positive even two-digit integers are there that do not contain the digits $0$, $1$, $2$, $3$ or $4$? [b]p6.[/b] In four years, Jack will be twice as old as Jill. Three years ago, Jack was three times as old as Jill. How old is Jack? [u]Round 3[/u] [b]p7.[/b] Let $x \Delta y = x y^2 -2y$. Compute $20\Delta 18$. [u]p8.[/u] A spider crawls $14$ feet up a wall. If Cheenu is standing $6$ feet from the wall, and is $6$ feet tall, how far must the spider jump to land on his head? [b]p9.[/b] There are fourteen dogs with long nails and twenty dogs with long fur. If there are thirty dogs in total, and three do not have long fur or long nails, how many dogs have both long hair and long nails? [u]Round 4[/u] [b]p10.[/b] Exactly $420$ non-overlapping square tiles, each $1$ inch by $1$ inch, tesselate a rectangle. What is the least possible number of inches in the perimeter of the rectangle? [b]p11.[/b] John drives $100$ miles at fifty miles per hour to see a cat. After he discovers that there was no cat, he drives back at a speed of twenty miles per hour. What was John’s average speed in the round trip? [b]p12.[/b] What percent of the numbers $1,2,3,...,1000$ are divisible by exactly one of the numbers $4$ and $5$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all citizens (including himself) make and wear a necklace with zero or more stones so that: (i) given a pair of friends, there exists a color such that each has a stone of that color; (ii) given a pair of enemies,there does not exist a color such that each a stone of that color. (a) Prove that the aborigines can carry out the chieftain’s order. (b) What is the minimum number of colors of stones required for the aborigines to carry out the chieftain’s order?

Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that [list] [*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal. [*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal. [*]There exist two cells in the grid that do not contain the same number. [/list] Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$. [i]Kiran Reddy[/i]

Let $n$ be a positive integer. Suppose that we have disks of radii $1, 2, . . . , n.$ Of each size there are two disks: a transparent one and an opaque one. In every disk there is a small hole in the centre, with which we can stack the disks using a vertical stick. We want to make stacks of disks that satisfy the following conditions: $i)$ Of each size exactly one disk lies in the stack. $ii)$ If we look at the stack from directly above, we can see the edges of all of the $n$ disks in the stack. (So if there is an opaque disk in the stack,no smaller disks may lie beneath it.) Determine the number of distinct stacks of disks satisfying these conditions. (Two stacks are distinct if they do not use the same set of disks, or, if they do use the same set of disks and the orders in which the disks occur are different.)

Let $A_{1}A_{2}\ldots A_{2n}$ be a convex polygon and let $P$ be a point in its interior such that it doesn't lie on any of the diagonals of the polygon. Prove that there is a side of the polygon such that none of the lines $PA_{1}$, $\ldots$, $PA_{2n}$ intersects it in its interior.

Natural numbers from $1$ to $200$ were divided into $50$ sets. Prove that one of them contains three numbers that are the lengths of the sides some triangle.

Let $n$ be a fixed positive integer. We have a $n\times n$ chessboard. We call a pair of cells [b]good[/b] if they share a common vertex (May be common edge or common vertex). How many [b]good[/b] pairs are there on this chessboard?

Submit an integer between $1$ and $50$, inclusive. You will receive a score as follows: If some number is submitted exactly once: If $E$ is your number, $A$ is the closest number to $E$ which received exactly one submission, and $M$ is the largest unique submission, you will receive $\frac{25}{M} (A - |E - A|)$ points, rounded to the nearest integer. If no number was submitted exactly once: If $E$ is your number, $K$ is the number of people who submitted $E$, and $M$ is the number of people who submitted the most commonly submitted number, then you will receive $\frac{25(M-K)}{M}$ points, rounded to the nearest integer.

Let $a$ and $b$ be two positive integers satifying $gcd(a, b) = 1$. Consider a pawn standing on the grid point $(x, y)$. A step of type A consists of moving the pawn to one of the following grid points: $(x+a, y+a),(x+a,y-a), (x-a, y + a)$ or $(x - a, y - a)$. A step of type B consists of moving the pawn to $(x + b,y + b),(x + b,y - b), (x - b,y + b)$ or $(x - b,y - b)$. Now put a pawn on $(0, 0)$. You can make a (nite) number of steps, alternatingly of type A and type B, starting with a step of type A. You can make an even or odd number of steps, i.e., the last step could be of either type A or type B. Determine the set of all grid points $(x,y)$ that you can reach with such a series of steps.