Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions: (i) Every groups contains an equal number of boys and girls. (ii) There is no connected pair in the same group. Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

At each node of the checkboard $n\times n$ board, a beetle sat. At midnight, each beetle crawled into the center of a cell. It turned out that the distance between any two beetles sitting in the adjacent (along the side) nodes didn't increase. Prove that at least one beetle crawled into the center of a cell at the vertex of which it sat initially. [i](A. Voidelevich)[/i]

In a board of $2000\times2001$ squares with integer coordinates $(x,y)$, $0\leq{x}\leq1999$ and $0\leq{y}\leq2000$. A ship in the table moves in the following way: before a move, the ship is in position $(x,y)$ and has a velocity of $(h,v)$ where $x,y,h,v$ are integers. The ship chooses new velocity $(h^\prime,v^\prime)$ such that $h^\prime-h,v^\prime-v\in\{-1,0,1\}$. The new position of the ship will be $(x^\prime,y^\prime)$ where $x^\prime$ is the remainder of the division of $x+h^\prime$ by $2000$ and $y^\prime$ is the remainder of the division of $y+v^\prime$ by $2001$. There are two ships on the board: The Martian ship and the Human trying to capture it. Initially each ship is in a different square and has velocity $(0,0)$. The Human is the first to move; thereafter they continue moving alternatively. Is there a strategy for the Human to capture the Martian, independent of the initial positions and the Martian’s moves? [i]Note[/i]: The Human catches the Martian ship by reaching the same position as the Martian ship after the same move.

Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.

We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.

There are seven points on the plane, no three of which are collinear. Every pair of points is connected with a line segment, each of which is either blue or red. Prove that there are at least four monochromatic triangles in the figure.

Determine the greatest possible value of n that satisfies the following condition: for any choice of $n$ subsets $M_1, ...,M_n$ of the set $M = \{1,2,...,n\}$ satisfying the conditions i) $i \in M_i$ and ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \ne j$, there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$. (Moscow regional olympiad,adopted)

There are $15$ lights on the ceiling of a room, numbered from $1$ to $15$. All lights are turned off. In another room, there are $15$ switches: a switch for lights $1$ and $2$, a switch for lights $2$ and $3$, a switch for lights $3$ en $4$, etcetera, including a sqitch for lights $15$ and $1$. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?

[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum. [b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img] [b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color. [b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points. (b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property. [b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$. (a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$. (b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k \leq 2022$ be a positive integer. Alice and Bob play a game on a $2022 \times 2022$ board. Initially, all cells are white. Alice starts and the players alternate. In her turn, Alice can either color one white cell in red or pass her turn. In his turn, Bob can either color a $k \times k$ square of white cells in blue or pass his turn. Once both players pass, the game ends and the person who colored more cells wins (a draw can occur). For each $1 \leq k \leq 2022$, determine which player (if any) has a winning strategy.

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.

When Ann meets new people, she tries to find out who is acquainted with who. In order to memorize it she draws a circle in which each person is depicted by a chord; moreover, chords corresponding to acquainted persons intersect (possibly at the ends), while the chords corresponding to non-acquainted persons do not. Ann believes that such set of chords exists for any company. Is her judgement correct? (5)

Determine the maximal length $L$ of a sequence $a_1,\dots,a_L$ of positive integers satisfying both the following properties: [list=disc] [*]every term in the sequence is less than or equal to $2^{2023}$, and [*]there does not exist a consecutive subsequence $a_i,a_{i+1},\dots,a_j$ (where $1\le i\le j\le L$) with a choice of signs $s_i,s_{i+1},\dots,s_j\in\{1,-1\}$ for which \[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\] [/list]

$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.

In a certain country there are more than $101$ towns. The capital of this country is connected by direct air routes with $100$ towns, and each town, except for the capital, is connected by direct air routes with $10$ towns (if $A$ is connected with $B, B$ is connected with $A$). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country. (IS Rubanov)

Consider a $ n \times n$ checkerboard with $ n > 1, n \in \mathbb{N}.$ How many possibilities are there to put $ 2n \minus{} 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the $ n \times n$ square.

Robert has two stacks of five cards numbered 1--5, one of which is randomly shuffled while the other is in numerical order. They pick one of the stacks at random and turn over the first three cards, seeing that they are 1, 2, and 3 respectively. What is the probability the next card is a 4? [i]Proposed by Connor Gordon[/i]

Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$

You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations: 1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta. 2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up. 3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively. 4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair. 5. Remove four consecutive beads of one color. Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'', b) ``cyan, yellow, cyan, magenta, cyan'', c) ``magenta, magenta, cyan, cyan, cyan'', d) ``yellow, cyan, cyan, cyan''. [i]Proposed by Sammy Luo[/i]

There are $100$ stones in a pile. A partition of the heap in $k $ piles is called [i]special [/i] if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the $k + 1$ piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number $k$, such that there is a special partition of the $100$ stones into $k $ piles. b) Find the minimum number $k $, such that there is a special partition of the $100$ stones in $k $ piles.

In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points. Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true: $$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.