There are some real numbers on the board (at least two). In every step we choose two of them, for example $a$ and $b$, and then we replace them with $\frac{ab}{a+b}$. We continue until there is one number. Prove that the last number does not depend on which order we choose the numbers to erase.

In a country, the time for presidential elections has approached. There are exactly 20 million voters in the country, of which only one percent supports the current president, Miraflores. Naturally, he wants to be elected again, but on the other hand, he wants the elections to seem democratic. Miraflores established the following voting process: all the voters are divided into several equal groups, then each of these groups is again divided into a number of equal groups, and so on. In the smallest groups, a representative is chosen. Then, the chosen electors choose representatives in the second-smallest groups, to vote in an even larger group, and so on. Finally, the representatives of the largest groups choose the president. Miraflores divides voters into groups as he wants and instructs his supporters how to vote. Will he be able to organize the elections in such a way that he will be elected president? (If the votes are equal, the opposition wins.) [i]From the 32nd Moscow Mathematical Olympiad[/i]

At the ends of a checkered strip measuring $1 \times 101$ squares there are two chips: on the left is the chip of the first player, on the right is the second. Per turn dares to move his piece in the direction of the opposite edge of the strip by 1, 2, 3 or 4 cells. In this case, you are allowed to jump over opponent's chip, but it is forbidden to place your chip on the same square with her. The first one to reach the opposite edge of the strip wins. Who wins if the game is played correctly: the one who goes first, or him rival?

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?

Given a positive integer $n$, determine the largest real number $\mu$ satisfying the following condition: for every set $C$ of $4n$ points in the interior of the unit square $U$, there exists a rectangle $T$ contained in $U$ such that $\bullet$ the sides of $T$ are parallel to the sides of $U$; $\bullet$ the interior of $T$ contains exactly one point of $C$; $\bullet$ the area of $T$ is at least $\mu$.

Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).

Compute the number of $10$-bit sequences of $0$’s and $1$’s do not contain $001$ as a subsequence.

Suppose $100$ points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?

Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.

Write down $1, 2, 3, ... , 2015$ in a row on a whiteboard. Every minute, select a pair of adjacent numbers at random, erase them, and insert their sum where you selected the numbers. (For instance, selecting $3$ and $4$ from $1, 2, 3, 4, 5$ would result in $1, 2, 7, 5$.) Repeat this process until you have two numbers remaining. What is the probability that the smaller number is less than or equal to $2015$?

An urn contains balls of $k$ different colors; there are $n_i$ balls of $i-th$ color. Balls are selected at random from the urn, one by one, without replacement, until among the selected balls $m$ balls of the same color appear. Find the greatest number of selections.

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$. a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisibel by $1408$.

All $n$-digit words from the alphabet $\{0, 1\}$ considered. These $2^n$ words should be in a sequence $w_0, w_1, w_2, ..., w_{2^-1}$ be arranged that $w_m$ from $w_{m-1}$ by changing of a single ornament ($m = 1, 2, 3, ..., 2n-1$). Prove that the following algorithm achievesthis : a) Start with $w_0 = 000... 00$. b) Let $w_{m-1} = a_1a_2a_3 ... a_n$ with $a_i \in \{0; 1\}$, $i = 1, 2, 3, ..., n$. Determine the exponent $e(m)$ of the highest power of two dividing $m$ and set $j = e(m)+1$. In $w_{m-1}$ replace the ornament $a_j$ with $1-aj$. this is now $w_m$.

On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

(a) On a blackboard are written $100$ different numbers. Prove that you can choose $8$ of them so that their average value is not equal to that of any $9$ of the numbers on the blackboard. (b) On a blackboard are written $100$ integers. For any $8$ of them, you can find $9$ numbers on the blackboard so that the average value of the $8$ numbers is equal to that of the $9$. Prove that all the numbers on the blackboard are equal. (A Shapovalov)

The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.

Let $k$ and $n$ be positive integers such that $k\le n-1$. Let $S:=\{1,2,\ldots,n\}$ and let $A_1,A_2,\ldots,A_k$ be nonempty subsets of $S$. Prove that it is possible to color some elements of $S$ using two colors, red and blue, such that the following conditions are satisfied: (i) Each element of $S$ is either left uncolored or is colored red or blue. (ii) At least one element of $S$ is colored. (iii) Each set $A_i~(i=1,2,\ldots,k)$ is either completely uncolored or it contains at least one red and at least one blue element.

The $m \times n$ chessboard is colored by black and white. In one step, two neighbouring squares are selected (squares with a common side) and their color changes according to the follwing way: - white becomes black, - black become red, - Red becomes white. For which $m$ and $n$, these steps can change the colors of all the initial squares from white to black and from black to white?

At a conference, six people place their name badges in a hat, which is shaken up; one badge is then distributed to each person such that each distribution is equally likely. Each turn, every person who does not yet have their own badge finds the person whose badge they have and takes that person's badge. For example, if Alice has Bob's badge and Bob has Charlie's badge, Alice would have Charlie's badge after a turn. Compute the probability that everyone will eventually end up with their own badge.

Given 2015 balls. Astri and Budi will play a game. At first, Astri will choose two different numbers $a$ and $b$ from the set $S = \{ 1, 2, 3, \dots, 30 \}$. Budi will then choose another 2 different numbers $c$ and $d$ from the remaining 28 numbers in set $S$. By taking turns, starting from Astri, they take balls with the following rules: (1) Astri could only take $a$ or $b$ balls. (2) Budi could only take $c$ or $d$ balls. until someone couldn't take any balls satisfying the condition given (and that person will lose). Prove that Budi could choose $c,d$ such that he has a strategy to ensure his victory on this game.

[u]Round 1[/u] [b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week? [b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img] [b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class? [img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img] [b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges. [img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img] [b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column? [u]Round 2[/u] [b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his? [img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img] [b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search. [img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].