A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$.

The numbers $1,2,\dots,2000$ are written on the board. Two players are playing a game with alternating moves. A move consists of erasing two number $a,b$ and writing $a^b$. After some time only one number is left. The first player wins, if the numbers last digit is $2$, $7$ or $8$. If not, the second player wins. Who has a winning strategy? [I]Proposed by V. Frank[/i]

[u]Set 4[/u] [b]p10.[/b] Eve has nine letter tiles: three $C$’s, three $M$’s, and three $W$’s. If she arranges them in a random order, what is the probability that the string “$CMWMC$” appears somewhere in the arrangement? [b]p11.[/b] Bethany’s Batteries sells two kinds of batteries: $C$ batteries for $\$4$ per package, and $D$ batteries for $\$7$ per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $\$2021$, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell? [b]p12.[/b] A deck of cards consists of $30$ cards labeled with the integers $1$ to $30$, inclusive. The cards numbered $1$ through $15$ are purple, and the cards numbered $16$ through $30$ are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number $32$. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck. PS. You should use hide for answers.

Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ chooses $x_1$ to be $1$ or $-1,$ then $B$ chooses $x_2$ to be $2$ or $-2,$ then $A$ chooses $x_3$ to be $3$ or $-3,$ and so on. (at the $k^{th}$ step, the chosen number must always be $k$ or $-k$) The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.

Evaluate $1 \oplus 2 \oplus \dots \oplus 987654321$ where $\oplus$ is bitwise exclusive OR. ($A\oplus B$ in binary has an $n$-th digit equal to $1$ if the $n$-th binary digits of $A$ and $B$ differ and $0$ otherwise. For example, $5 \oplus 9 = 0101_{2} \oplus 1001_{2} = 1100_2= 12$ and $6 \oplus 7 = 110_2 + 111_2 = 001_2 = 1$.) [i]Proposed by Jacob Weiner[/i]

Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\).

Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?

The $n$ players of a hockey team gather to select their team captain. Initially, they stand in a circle, and each person votes for the person on their left. The players will update their votes via a series of rounds. In one round, each player $a$ updates their vote, one at a time, according to the following procedure: At the time of the update, if $a$ is voting for $b,$ and $b$ is voting for $c,$ then $a$ updates their vote to $c.$ (Note that $a, b,$ and $c$ need not be distinct; if $b=c$ then $a$'s vote does not change for this update.) Every player updates their vote exactly once in each round, in an order determined by the players (possibly different across different rounds). They repeat this updating procedure for $n$ rounds. Prove that at this time, all $n$ players will unanimously vote for the same person.

Santa Claus plays a guessing game with Marvin before giving him his present. He hides the present behind one of $100$ doors, numbered from $1$ to $100$. Marvin can point at a door, and then Santa Claus will reply with one of the following words: $\bullet$ "hot" if the present lies behind the guessed door, $\bullet$ "warm" if the guess is not exact but the number of the guessed door differs from that of the present’s door by at most $5$, $\bullet$ "cold" if the numbers of the two doors differ by more than $5$. At least how many such guesses does Marvin need, so that he can be certain about where his present is? Marvin does not necessarily need to make a "hot" guess, just to know the correct door with $100\%$ certainty.

Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it

Some cells of an infinite chessboard (infinite in all directions) are coloured blue so that at least one of the $100$ cells in any $10 \times 10$ rectangular grid is blue. Prove that, for any positive integer $n$, it is possible to select $n$ rows and $n$ columns so that all of the $n^2$ cells in their intersections are blue.

Let there be a finite number of straight lines in the plane, none of which are three in one point to cut. Show that the intersections of these straight lines can be colored with $3$ colors so that that no two points of the same color are adjacent on any of the straight lines. (Two points of intersection are called [i]adjacent [/i] if they both lie on one of the finitely many straight lines and there is no other such intersection on their connecting line.)

Two teams $A$ and $B$ play a school ping pong tournament. The team $A$ consists of $m$ students, and the team $B$ consists of $n$ students where $m \ne n$. There is only one ping pong table to play and the tournament is organized as follows: Two students from different teams start to play while other players form a line waiting for their turn to play. After each game the first player in the line replaces the member of the same team at the table and plays with the remaining player. The replaced player then goes to the end of the line. Prove that every two players from the opposite teams will eventually play against each other.

Suppose that $1000$ students are standing in a circle. Prove that there exists an integer $k$ with $100 \leq k \leq 300$ such that in this circle there exists a contiguous group of $2k$ students, for which the first half contains the same number of girls as the second half. [i]Proposed by Gerhard Wöginger, Austria[/i]

Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

There are three types of tiles: an L-shaped tile with three $1\times 1$ squares, a $2\times 2$ square, and a Z-shaped tile with four $1\times 1$ squares. We tile a $(2n-1)\times (2n-1)$ square using these tiles. Prove that there are at least $4n-1$ L-shaped tiles. I'm sorry about my poor description, but I don't know how to draw pictures...

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

Let $a_1,a_2,...,a_n$ be pairwise distinct real numbers and $m$ be the number of distinct sums $a_i +a_j$ (where $i \ne j$). Find the least possible value of $m$.

[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets. (a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket. (b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket. [b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties. (a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$. (b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$. [b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer. [b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$. [b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that: (a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle. (b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.

Given numbers $1, 2, . . .,N$, each of which is colored either black or white. It is allowed to repaint it in the opposite direction color any three numbers, one of which is equal to half the sum of the other two. At which $N$ numbers can always be made white?

Prove that there exists a positive integer $k>100$, such that for any set $A$ of $k$ positive reals, there exists a subset $B$ of $100$ numbers, so that none of the sums of at least two numbers in $B$ is in the set $A$.