In a party attended by $2015$ guests among any $5$ guests at most $6$ handshakes had been exchanged. Determine the maximal possible total number of handshakes.

On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points. $a)$ Prove that there exist two students with exactly the same answers to problems $b)$ Prove that there exist at least $4$ students with same number of points

There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?

Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint [i]regions[/i]. Find the number of such regions.

When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$? $\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$

The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

Let $n> 1$ be a positive integer. Danielle chooses a number $N$ of $n$ digits but does not tell her students and they must find the sum of the digits of $N$. To achieve this, each student chooses and says once a number of $n$ digits to Danielle and she tells how many digits are in the correct location compared with $N$. Find the minimum number of students that must be in the class to ensure that students have a strategy to correctly find the sum of the digits of $N$ in any case and show a strategy in that case.

Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser? [img]http://i38.tinypic.com/2i1icec_th.png[/img] [img]http://i34.tinypic.com/28rk4s3_th.png[/img] a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime. b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser. Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.

Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j \minus{} x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.

Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.

Milo rolls five fair dice which have 4, 6, 8, 12, and 20 sides respectively (and each one is labeled $1$-$n$ for appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1,1,2,2)$ and $(2,2,1,1,1)$ would both be counted. [i]Proposed by Robert Trosten[/i]

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

On an $m\times n$ grid there's a snail in each cell. Each round, the snail army can choose four snail in a $2\times 2$ square and make them perform the complete [b]Quadrilateral Dance[/b], which is rotating the four snails clockwise by $90$ degrees, moving each one to an adjacent cell. Find all pairs of positive integers $(m,n)$ such that the snails can achieve any permutation by performing a finite number of times of [b]Quadrilateral Dance[/b]. [i]Proposed by chengbilly[/i]

[u]Round 1[/u] [b]p1.[/b] Alice and Bob compiled a list of movies that exactly one of them saw, then Cindy and Dale did the same. To their surprise, these two lists were identical. Prove that if Alice and Cindy list all movies that exactly one of them saw, this list will be identical to the one for Bob and Dale. [b]p2.[/b] Several whole rounds of cheese were stored in a pantry. One night some rats sneaked in and consumed $10$ of the rounds, each rat eating an equal portion. Some were satisfied, but $7$ greedy rats returned the next night to finish the remaining rounds. Their portions on the second night happened to be half as large as on the first night. How many rounds of cheese were initially in the pantry? [b]p3.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake, and call that number N. Pick up the stack of the top N pancakes, and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack. [b]p4.[/b] There are two lemonade stands along the $4$-mile-long circular road that surrounds Sour Lake. $100$ children live in houses along the road. Every day, each child buys a glass of lemonade from the stand that is closest to her house, as long as she does not have to walk more than one mile along the road to get there. A stand's [u]advantage [/u] is the difference between the number of glasses it sells and the number of glasses its competitor sells. The stands are positioned such that neither stand can increase its advantage by moving to a new location, if the other stand stays still. What is the maximum number of kids who can't buy lemonade (because both stands are too far away)? [b]p5.[/b] Merlin uses several spells to move around his $64$-room castle. When Merlin casts a spell in a room, he ends up in a different room of the castle. Where he ends up only depends on the room where he cast the spell and which spell he cast. The castle has the following magic property: if a sequence of spells brings Merlin from some room $A$ back to room $A$, then from any other room $B$ in the castle, that same sequence brings Merlin back to room $B$. Prove that there are two different rooms $X$ and $Y$ and a sequence of spells that both takes Merlin from $X$ to $Y$ and from $Y$ to $X$. [u]Round 2[/u] [b]p6.[/b] Captains Hook, Line, and Sinker are deciding where to hide their treasure. It is currently buried at the $X$ in the map below, near the lairs of the three pirates. Each pirate would prefer that the treasure be located as close to his own lair as possible. You are allowed to propose a new location for the treasure to the pirates. If at least two out of the three pirates prefer the new location (because it moves closer to their own lairs), then the treasure will be moved there. Assuming the pirates’ lairs form an acute triangle, is it always possible to propose a sequence of new locations so that the treasure eventually ends up in your backyard (wherever that is)? [img]https://cdn.artofproblemsolving.com/attachments/c/c/a9e65624d97dec612ef06f8b30be5540cfc362.png[/img] [b]p7.[/b] Homer went on a Donut Diet for the month of May ($31$ days). He ate at least one donut every day of the month. However, over any stretch of $7$ consecutive days, he did not eat more than $13$ donuts. Prove that there was some stretch of consecutive days over which Homer ate exactly $30$ donuts. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.

For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.

In a chess tournament, each of $2n$ players plays every other player once in each of two rounds. A win is worth $1$ point, a draw is worth $\frac12$ point and a loss is worth nothing. Prove that if for every player, the total score in the first round differs from that in the second round by at least n points, then this difference is exactly n points for every player. (B Frenkin)

Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.

At CMIMC headquarters, there is a row of $n$ lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things: [list] [*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. [/list] Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most $n$ seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value $n$ such that he can complete his task on time? [i]Proposed by Adam Bertelli[/i]

Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell [i]correct [/i] if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be [i]correct[/i]?

Let \(\mathbb{Z}^2\) denote the set of points in the Euclidean plane with integer coordinates. Find all functions \(f : \mathbb{Z}^2 \to [0,1]\) such that for any point \(P\), the value assigned to \(P\) is the average of all the values assigned to points in \(\mathbb{Z}^2\) whose Euclidean distance from \(P\) is exactly 2020.

A SET cards have four characteristics: number, color, shape, and shading, each of which has $3$ values. A SET deck has $81$ cards, one for each combination of these values. A SET is three cards such that, for each characteristic, the values of the three cards for that characteristics are either all the same or all different. In how many ways can you replace each SET card in the deck with another SET card (possibly the same), with no card used twice, such that any three cards that were a SET before are still a SET? (Alternately, a SET card is an ordered $4$-tuple of $0$s, $1$s, and $2$s, and three cards form a SET if their sum is ($0, 0, 0, 0$) mod $3$, for instance, ($0, 1, 2, 2$), ($1, 0, 2, 1$), and ($2, 2, 2, 0$) form a SET. How many permutations of the SET cards maintain SET-ness?)

N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$. I.Gorodnin

There are $33$ children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers $0,1,2,\dots,10$ appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.

Starting at $(1,1)$, a stone is moved in the coordinate plane according to the following rules: (i) From any point $(a,b)$, the stone can move to $(2a,b)$ or $(a,2b)$. (ii) From any point $(a,b)$, the stone can move to $(a-b,b)$ if $a > b$, or to $(a,b-a)$ if $a < b$. For which positive integers $x,y$ can the stone be moved to $(x,y)$?