Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four. (a) Write down an arrangement for a tournament of $24$ games satisfying these conditions. (b) Show that it is impossible to have a tournament of more than $24$ games satisfying these conditions.

Frodo composes a number triangle of zeroes and ones in such a way: he fills the topmost row with any $n$ digits, and in other rows he always writes $0$ under consecutive equal digits and writes $1$ under consecutive distinct digits. (An example of a triangle for $n=5$ is shown below.) In how many ways can Frodo fill the topmost row for $n=100$ so that each of $n$ rows of the triangle contains odd number of ones?\[\begin{smallmatrix}1\,0\,1\,1\,0\\1\,1\,0\,1\\0\,1\,1\\1\,0\\1\end{smallmatrix}\] (O. Rudenko and V. Brayman)

A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be [i]dispersed [/i]i f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is [i]dispersed[/i], but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be [i]correctable[/i]. Indicates all integers $n$ for which there is a network with $n$ stations, [i]dispersed [/i]and not [i]correctable[/i].

Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$

Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. [i]F. Petrov [/i]

Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.

[b]p1.[/b] Insert "$+$" signs between some of the digits in the following sequence to obtain correct equality: $$1\,\,\,\, 2\,\,\,\, 3\,\,\,\, 4\,\,\,\,5\,\,\,\, 6\,\,\,\, 7 = 100$$ [b]p2.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the big square $ABCD$ is $40$ cm. [img]https://cdn.artofproblemsolving.com/attachments/8/c/d54925cba07f63ec8578048f46e1e730cb8df3.png[/img] [b]p3.[/b] Jack made $3$ quarts of fruit drink from orange and apple juice. $\frac25$ of his drink is orange juice and the rest is apple juice. Nick prefers more orange juice in the drink. How much orange juice should he add to the drink to obtain a drink composed of $\frac35$ of orange juice? [b]p4.[/b] A train moving at $55$ miles per hour meets and is passed by a train moving moving in the opposite direction at $35$ miles per hour. A passenger in the first train sees that the second train takes $8$ seconds to pass him. How long is the second train? [b]p5.[/b] It is easy to arrange $16$ checkers in $10$ rows of $4$ checkers each, but harder to arrange $9$ checkers in $10$ rows of $3$ checkers each. Do both. [b]p6.[/b] Every human that lived on Earth exchanged some number of handshakes with other humans. Show that the number of people that made an odd number of handshakes is even. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$. We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$). A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd). (a)Show that the number of even sets equals the number of odd sets. (b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets. (c)Compute the sum of the measures of the odd sets.

A positive integer is said to be a “palindrome” if it reads the same from left to right as from right to left. For example 2002 is a palindrome. Find the sum of all 4-digit palindromes.

If $\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}$ be vectors in the plane such that the sum of their lengths is at least $1$, then between them we find vectors whose sum is a vector of length at least $\sqrt2/8$. Prove it.

Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).

A table consists of $17 \times 17$ squares. In each square one positive integer from $1$ to $17$ is written, every such number is written in exactly $17$ squares. Prove that there is a row or a column of the table that contains at least $5$ different numbers.

We are given a set $S=\{z_1,z_2,\cdots ,z_{1993}\}$, where $z_1,z_2,\cdots ,z_{1993}$ are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide $S$ into some groups such that the following conditions are satisfied: (1) Each element in $S$ belongs and only belongs to one group; (2) For any group $p$, if we use $T(p)$ to denote the sum of all memebers in $p$, then for any memeber $z_i (1\le i \le 1993)$ of $p$, the angle between $z_i$ and $T(p)$ does not exceed $90^{\circ}$; (3) For any two groups $p$ and $q$, the angle between $T(p)$ and $T(q)$ exceeds $90^{\circ}$ (use the notation introduced in (2)).

Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time. Find the number of all attainable tables.

There are $2n+1$ segments on the line. Any segment intersects at with at least $n$ others. Prove that there is a segment that intersects all the others.

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

Given a positive integer $n\ge 3$, Arándano and Banana play a game. Initially, numbers $1,2,3,\dots,n$ are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise. Determine, in terms of $n$, who has a winning strategy.

A set of $n^2$ counters are labeled with $1,2,\ldots, n$, each label appearing $n$ times. Can one arrange the counters on a line in such a way that for all $x \in \{1,2,\ldots, n\}$, between any two successive counters with the label $x$ there are exactly $x$ counters (with labels different from $x$)?

A tourist wants to visit each of the ten cities shown on the picture. The continuous segments on the picture denote railway lines, whereas the dashed segments denote air lines. A railway line costs $150000$ lires, and an air line costs $250000$ lires. What is the minimum possible price of a desired route? [asy] unitsize(2.5 cm); real r = 0.05; pair A, B, C, D, E, F, G, H, I, J; A = (0,0); B = dir(30); D = dir(-30); C = B + D; E = (A + B)/2; F = (B + C)/2; G = (C + D)/2; H = (D + A)/2; I = (A + B + D)/3; J = (B + C + D)/3; draw(A--B--C--D--cycle); draw(E--I--H); draw(F--J--G); draw(B--D, dashed); draw(E--H, dashed); draw(F--G, dashed); draw(I--J, dashed); filldraw(Circle(A,r),white); filldraw(Circle(B,r),white); filldraw(Circle(C,r),white); filldraw(Circle(D,r),white); filldraw(Circle(E,r),white); filldraw(Circle(F,r),white); filldraw(Circle(G,r),white); filldraw(Circle(H,r),white); filldraw(Circle(I,r),white); filldraw(Circle(J,r),white); label("$A$", A + r*dir(225), SW); [/asy]

Find the number of subsets of the set $\{1,2,3,...,5n\}$ such that the sum of the elements in each subset are divisible by $5$.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability $\frac13$ , left with probability $\frac13$ , and right with probability $\frac13$. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is $\frac{p}{q}$ , where $p$ and $q$ are coprime positive integers, then your answer should be $p + q$. [i]The steps of the ant are independent.[/i]

Andryusha has $100$ stones of different weight and he can distinguish the stones by appearance, but does not know their weight. Every evening, Andryusha can put exactly $10$ stones on the table and at night the brownie will order them in increasing weight. But, if the drum also lives in the house then surely he will in the morning change the places of some $2$ stones.Andryusha knows all about this but does not know if there is a drum in his house. Can he find out?

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.

$22$ light bulbs are given. Each light bulb is connected to exactly one switch, but a switch can be connected to one or more light bulbs. Find the least number of switches we should have such that we can turn on whatever number of light bulbs.