Prove that in a meeting of $285$ people, at least one of them has given an even number of handshakes ($0$ is considered an even number and corresponds to an assistant who does not shake any hand).

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

A grandpa has a finite number of boxes in his attic. Each box is a straight rectangular prism with integer edge lengths. For every box its width is greater or equal to its height and its length is greater or equal to its width. A box can be put inside another box if and only if all of its dimensions are respectively smaller than the other one's. You can put two or more boxes in a bigger box only if the smaller boxes are all already inside one of the boxes. The grandpa decided to put the boxes in each other so that there would be a minimal number of visible boxes in the attic (boxes that have not been put inside another). He decided to use the following algorithm: at each step he finds the longest sequence of boxes so that the first can be put in the second, the second can be put in the third, etc., and then he puts them inside each other in the aforementioned order. The grandpa used the algorithm until no box could be put inside another. It is known that at each step the longest sequence of boxes was unique, e.g., at no moment were there two different sequences with the same length. The grandpa now claims that he has the minimal possible number of visible boxes in his attic. Is the claim necessarily true?

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour? (Mebius, Sharygin)

A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way. [img]https://cdn.artofproblemsolving.com/attachments/e/4/f7a04300f0c846fb6418d12dc23f5c74b54242.png[/img]

Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

There are $n$ (an even number) bags. Each bag contains atleast one apple and at most $n$ apples. The total number of apples is $2n$. Prove that it is always possible to divide the bags into two parts such that the number of apples in each part is $n$.

We have a set of 2n positive integers whose sum is a multiple of n. One operation consists of choosing n of them and adding the same positive integer to all of them. Show that, starting from the initial 2n numbers, we can get all are equal, performing a maximum of 2n - 1 operations.

There are $n$ assassins numbered from $1$ to $n,$ and all assasins are initially alive. The assassins play a game in which they take turns in increasing order of number, with assassin $1$ getting the first turn, then assassin $2$, etc., with the order repeating after assassin $n$ has gone; if an assassin is dead when their turn comes up, then their turn is skipped and it goes to the next assassin in line. On each assassin’s turn, they can choose to either kill the assassin who would otherwise move next or to do nothing. Each assassin will kill on their turn unless the only option for guaranteeing their own survival is to do nothing. If there are $2023$ assassins at the start of the game, after an entire round of turns in which no one kills, how many assassins must remain?

Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously: (1) the sum of numbers in every column is one, (2) the sum of every two neighboring numbers in the first row is at most one, (3) the sum of every two neighboring numbers in the second row is at most one, (4) the sum of every two neighboring numbers in the third row is at most $\alpha$.

Let $n$ cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card. (a)If $n=2015$ , can you make all cards white-side-up through a finite number of moves? (b)If $n=2016$ , can you make all cards white-side-up through a finite number of moves?

Let $n$ be the number of ordered $5$-tuples $(a_1,a_2,\ldots,a_5)$ of positive integers such that $\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_5}=1$. Is $n$ an even number?

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

On every card of a deck of cards a regular 17-gon is displayed with all sides and diagonals, and the vertices are numbered from 1 through 17. On every card all edges (sides and diagonals) are colored with a color 1,2,...,105 such that the following property holds: for every 15 vertices of the 17-gon the 105 edges connecting these vertices are colored with different colors on at least one of the cards. What is the minimum number of cards in the deck?

Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$. For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$-balanced.(Czech Republic)

[b]p1.[/b] Solve for $x$ and $y$ if $\frac{1}{x^2}+\frac{1}{xy}=\frac{1}{9}$ and $\frac{1}{y^2}+\frac{1}{xy}=\frac{1}{16}$ [b]p2.[/b] Find positive integers $p$ and $q$, with $q$ as small as possible, such that $\frac{7}{10} <\frac{p}{q} <\frac{11}{15}$. [b]p3.[/b] Define $a_1 = 2$ and $a_{n+1} = a^2_n -a_n + 1$ for all positive integers $n$. If $i > j$, prove that $a_i$ and $a_j$ have no common prime factor. [b]p4.[/b] A number of points are given in the interior of a triangle. Connect these points, as well as the vertices of the triangle, by segments that do not cross each other until the interior is subdivided into smaller disjoint regions that are all triangles. It is required that each of the givien points is always a vertex of any triangle containing it. Prove that the number of these smaller triangular regions is always odd. [b]p5.[/b] In triangle $ABC$, let $\angle ABC=\angle ACB=40^o$ is extended to $D$ such that $AD=BC$. Prove that $\angle BCD=10^o$. [img]https://cdn.artofproblemsolving.com/attachments/6/c/8abfbf0dc38b76f017b12fa3ec040849e7b2cd.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A $3 \times 3 \times 4$ cuboid is constructed out of 36 white-coloured unit cubes. Then, all six of the cuboid's sides are coloured red. After that, the cuboid is dismantled into its constituent unit cubes. Then, randomly, all said unit cubes are constructed into the cuboid of its original size (and position). a) How many ways are there to position eight of its corner cubes so that the apparent sides of eight corner cubes are still red? (Cube rotations are still considered distinct configurations, and the position of the cuboid remains unchanged.) b) Determine the probability that after the reconstruction, all of its apparent sides are still red-coloured. (The cuboid is still upright, with the same dimensions as the original cuboid, without rotation.) [hide=Notes]The problem might have multiple interpretations. We agreed that this problem's wording was a bit ambiguous. Here's the original Indonesian version: Suatu balok berukuran $3 \times 3 \times 4$ tersusun dari 36 kubus satuan berwarna putih. Kemudian keenam permukaan balok diwarnai merah. Setelah itu, balok yang tersusun dari kubus-kubus satuan tersebut dibongkar. Kemudian, secara acak, semua kubus satuan disusun lagi menjadi balok seperti balok semula. a) Ada berapa cara menempatkan kedelapan kubus satuan yang berasal dari pojok sehingga kedelapan kubus di pojok yang tampak tetap berwarna merah? (Rotasi kubus dianggap konfigurasi yang berbeda, namun posisi balok tidak diubah.) b) Tentukan probabilitas balok yang tersusun lagi semua permukaannya berwarna merah. (Balok tegak tetap tegak dan balok tetap dalam suatu posisi.) [/hide]

Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.

Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.

Prove it is possible to find $2^{2021}$ different pairs of positive integers $(a_i,b_i)$ such that: $$ \frac{1}{a_ib_i}+\frac{1}{a_2b_2} + \ldots + \frac{1}{a_{2^{2021}}b_{2^{2021}}} = 1 $$ $$ a_1+a_2 +\ldots a_{2^{2021}} +b_1+b_2 + \ldots +b_{2^{2021}} = 3^{2022} $$ [b]Note: [/b]Pairs $(a,b)$ and $(c,d)$ are different if $a \neq c$ or $b \neq d$

A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

We have $2n$ lights in two rows, numbered from $1$ to $n$ in each row. Some (or none) of the lights are on and the others are off, we call that a "state". Two states are distinct if there is a light which is on in one of them and off in the other. We say that a state is good if there is the same number of lights turned on in the first row and in the second row. Prove that the total number of good states divided by the total number of states is: $$ \frac{3 \cdot 5 \cdot 7 \cdots (2n-1)}{2^n n!} $$

Every number in the sequence $1, 2, ... , 2021$ is either white or black. At one step Alice can choose three numbers of the sequence and change the color of each of them (white to black and black to white) if one of those three numbers is the arithmetic mean of the other two. Alice wants to perform several steps so that at the end all the numbers in the sequence are black. For which initial colorings of numbers can Alice achieve this?