Mari and Yuri play the next play. At first, there are two piles on the table, with $m$ and $n$ candies, respectively. At each turn, players eats one pile of candy from the table and distribute another pile of candy into two non-empty parts ,. Everything is done in turn and wins the player who can no longer share the pile (when there is only one candy left). Which player will win if both use the optimal strategy and Mari makes the first move?

A connected checkered figure is drawn on a checkered paper. It is known that the figure can be cut both into $2\times 2$ squares and into (possibly rotated) [url=https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tetromino-skew2.svg/1200px-Tetromino-skew2.svg.png]skew-tetrominoes[/url]. Prove that there is a hole in the figure. [i]Proposed by Y. Markelov and A. Sairanov[/i]

There are some cities in both sides of a river and there are some sailing channels between the cities. Each sailing channel connects exactly one city from a side of the river to a city on the other side. Each city has exactly $k$ sailing channels. For every two cities, there's a way which connects them together. Prove that if we remove any (just one) sailing channel, then again for every two cities, there's a way that connect them together. $( k \geq 2)$

Let $G$ be a simple connected graph with $2016$ vertices and $k$ edges. We want to choose a set of vertices where there is no edge between them and delete all these chosen vertices (we delete both the vertices and all edges of these vertices) such that the remaining graph becomes unconnected. If we can do this task no matter how these $k$ edges are arranged (by making the graph connected), find the maximal value of $k$.

In the figure, the rectangle is formed by $4$ smaller equal rectangles. If we count the total number of rectangles in the figure we find $10$. How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

All internal angles of a convex octagon $ABCDEFGH$ are equal to each other and the edges are alternatively equal: $$AB = CD = EF = GH,BC = DE = FG = HA$$ (we call such an octagon semiregular). The diagonals $AD$, $BE$, $CF$, $DG$, $EH$, $FA$, $GB$ and $HC$ divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

On the board all the 20-digit numbers which have 10 ones and 10 twos in their decimal form are written. It is allowed to choose two different digits in any number and to reverse the order of digits in the interval between them. What is the maximal quantity of equal numbers which is possible to get on the board using such operations?

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements: [list] [*] There are no more than three of us here. We are all liars. [*] There are no more than four of us here. Not all of us are liars. [*] There are five of us here. At least three of us are liars. [/list] How many people are in the room and how many of them are liars?

There are 8 coins, 7 of which are real, which weigh the same, and one is fake, which differs in weight from the rest. Cup scales without weights mean that if you put equal weights on their cups, then any of the cups can outweigh, but if the loads are different in mass, then the cup with a heavier load is definitely overpowered. How to definitely identify a counterfeit coin in four weighings and establish is it lighter or heavier than the others?

Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy] unitsize(.85cm); draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1)); draw((0,2)--(0,1)--(1,1)--(1,2),dashed); label("\footnotesize $a$",(0.5,0),S); label("\footnotesize $a$",(0,0.5),W); label("\footnotesize $a$",(1,0.5),E); label("\footnotesize $a$",(0,1.5),E); label("\footnotesize $a$",(1,1.5),W); label("\footnotesize $a$",(-0.5,2),N); label("\footnotesize $a$",(0.5,2),N); label("\footnotesize $a$",(1.5,2),N); [/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length $a$ of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of $\lambda$ such that every such configuration of tiles has a total area less than $\lambda$.

It is election year in PUMACland, and for the presidential election there are $27$ people voting for either Vraj Patel or Vedant Shah. Each voter selects a candidate uniformly at random, and their ballots are labeled $1$ through $27.$ The election takes place as a series of rounds. In each round, the surviving ballots are sorted by label and separated into consecutive groups of three. From each group, the person with the most votes wins, and exactly one of the ballots bearing the winner’s name is allowed to proceed to the next round. This procedure continues until a single ballot remains, and the person whose name is on the ballot wins. Alice, Bob, and Carol submitted ballots numbered $1, 15,$ and $27,$ respectively. Suppose that Alice, Bob, and Carol had all flipped their votes. If the probability that the outcome of the election would have changed is $\tfrac{a}{b}$ for relatively prime positive integers $a, b,$ find $a + b.$

Let $a_1,a_2,...$ be a sequence of non-negative integers such that for any $m,n$ \[ \sum_{i=1}^{2m} a_{in} \leq m.\] Show that there exist $k,d$ such that \[ \sum_{i=1}^{2k} a_{id} = k-2014.\]

There are $n$ people in line, counting $1,2,\cdots, n$ from left to right, those who count odd numbers quit the line, the remaining people press 1,2 from right to left, and count off again, those who count odd numbers quit the line, and then the remaining people count off again from left to right$\cdots$ Keep doing that until only one person is in the line. $f(n)$ is the number of the last person left at the first count. Find the expression for $f(n)$ and find the value of $f(2022)$

There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

a) Several black squares of side $1$ cm are nailed to a white plane with a nail of thickness $0 . 1$ cm so that they form a black polygon. Can it happen that the perimeter of this polygon is $1$ km long? (The nail is not allowed to touch the boundary of any of the squares . ) (b) The same problem as in (a) but with a nail of thickness $0$ (a point ) . (c) Several black squares of side $1$ cm lie on a white plane so that they form a black polygon (possibly having more than one piece and/ or having holes) . Can it happen that the ratio of its perimeter (in centimetres) to its area (in square centimetres) is more than $100000$? (Hungarian Folklore)

Let be two natural numbers $ m\ge n $ and a nonnegative integer $ r<2^n. $ How many numbers of $ m $ digits, each digit being either the number $ 1 $ or $ 2, $ are there whose residue modulo $ 2^n $ is $ r? $ [i]Dorel Miheț[/i]

Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.

Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal:\\ (1) $f$ has periodic $2017$\\ (2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$\\ Proposed by William Chao

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

Given several matrices of the same size. Given a positive integer $N$, let's say that a matrix is $N$-balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$. (i) Show that every $2N$-balanced matrix can be written as a sum of two $N$-balanced matrices. (ii) Show that every $3N$-balanced matrix can be written as a sum of three $N$-balanced matrices.

There are $30$ contestants and each contestant has $6$ friends each. $3$ people is selected from these $30$ contestants, and it is called $good~triple$, if either all three are mutual friends, or none of them are friends with each other. How many $good~triples$ are there? (Note: If contestant $A$ is friends with $B$, then $B$ is friends with $A$. Similarly, if $A$ is not friends with $B$, then $B$ is not friends with $A$)

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.

Martin has two boxes $A$ and $B$. In the box $A$ there are $100$ red balls numbered from $1$ to $100$, each one with one of these numbers. In the box $B$ there are $100$ blue balls numbered from $101$ to $200$, each one with one of these numbers. Martin chooses two positive integers $a$ and $b$, both less than or equal to $100$, and then he takes out $a$ balls from box $A$ and $b$ balls from box $B$, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.\\ What is the least possible value of $a+b$ so that Martin achieves his goal for sure? For such a minimum value of $a+b$, give an example of $a$ and $b$ satisfying the goal and explain why every $a$ and $b$ with smaller sum cannot accomplish the aim.