There are three types of piece shown as below. Today Alice wants to cover a $100 \times 101$ board with these pieces without gaps and overlaps. Determine the minimum number of $1\times 1$ pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.) [asy] size(9cm,0); defaultpen(fontsize(12pt)); draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle); draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle); draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle); draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle); draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle); draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle); draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle); draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle); draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle); draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle); draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle); draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle); draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle); draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle); [/asy] [i]Proposed by amano_hina[/i]

[b]p1.[/b] One day, Granny Smith bought a certain number of apples at Horock’s Farm Market. When she returned the next day she found that the price of the apples was reduced by $20\%$. She could therefore buy more apples while spending the same amount as the previous day. How many percent more? [b]p2.[/b] You are asked to move several boxes. You know nothing about the boxes except that each box weighs no more than $10$ tons and their total weight is $100$ tons. You can rent several trucks, each of which can carry no more than $30$ tons. What is the minimal number of trucks you can rent and be sure you will be able to carry all the boxes at once? [b]p3.[/b] The five numbers $1, 2, 3, 4, 5$ are written on a piece of paper. You can select two numbers and increase them by $1$. Then you can again select two numbers and increase those by $1$. You can repeat this operation as many times as you wish. Is it possible to make all numbers equal? [b]p4.[/b] There are $15$ people in the room. Some of them are friends with others. Prove that there is a person who has an even number of friends in the room. [u]Bonus Problem [/u] [b]p5.[/b] Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?

Find maximal numbers of planes, such there are $6$ points and 1) $4$ or more points lies on every plane. 2) No one line passes through $4$ points.

Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B,A,B,A,\ldots $ by the following rules: [b](a)[/b] On every move of his $B$ passes $1$ coin from every box to an adjacent box. [b](b)[/b] On every move of hers $A$ chooses several coins that were [i]not[/i] involved in $B$'s previous move and are in different boxes. She passes every coin to an adjacent box. Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

When flipped, a coin has a probability $p$ of landing heads. When flipped twice, it is twice as likely to land on the same side both times as it is to land on each side once. What is the larger possible value of $p$?

There are some gas stations on a circular highway. The total amount of gasoline in them is enough for two laps. Two drivers want to refuel at one station and starting from it, go in different directions, both of them completing an entire lap. Along the way, they can refuel at other stations, without necessarily taking all the gasoline. Prove that drivers will always be able to do this. [i]Proposed by I. Bogdanov[/i]

Fix a positive integer $n$ and a finite graph with at least one edge; the endpoints of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range from $0$ to $n-1$, one number each. A vertex assignment and an edge assignment are [i]compatible[/i] if the following condition is satisfied at each vertex $v$: The number assigned to $v$ is congruent modulo $n$ to the sum of the numbers assigned to the edges incident to $v$. Fix a vertex assignment and let $N$ be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment. Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$.

Bob plotted $2003$ green points on the plane, so all triangles with three green vertices have area less than $1$. Prove that the $2003$ green points are contained in a triangle $T$ of area less than $4$.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

The King gives the following task to his two wizards. The First Wizard should choose $7$ distinct positive integers with total sum $100$ and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand. (Mikhail Evdokimov)

There are $n$ points in the plane, some of which are connected by segments. Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the $n$ points exactly once. It is known that the segments $AB$ and $BC$ are white. Prove that it is possible to recolor the segments in red and blue in such a way that $AB$ and $BC$ are recolored as red, [hide=not all of which segments are recolored red]meaning that recoloring every white as red and every black as blue is not acceptable[/hide], and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the $n$ points exactly once.

In a board, the positive integer $N$ is written. In each round, Olive can realize any one of the following operations: I - Switch the current number by a positive multiple of the current number. II - Switch the current number by a number with the same digits of the current number, but the digits are written in another order(leading zeros are allowed). For instance, if the current number is $2022$, Olive can write any of the following numbers $222,2202,2220$. Determine all the positive integers $N$, such that, Olive can write the number $1$ after a finite quantity of rounds.

A square of size $2\times2$ with one of its cells occupied by a tower is called a castle. What maximal number of castles one can place on a board of size $7\times7$ so that the castles have no common cells and all the towers stand on the diagonals of the board?

In each box of a $9\times 9$ grid we write a positive integer such that, between any $2$ boxes on the same row or column that have the same number $n$ written, there's at least $n$ boxes between them. What is the minimum sum possible for the numbers on the grid?

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

You are given $25$ pieces of cheese of different weights. Is it always possible to cut one of the pieces into two parts and put the $26$ pieces in two packets so that $\bullet$ each packet contains $13$ pieces; $\bullet$ the total weights of the two packets are equal; $\bullet$ the two parts of the piece which has been cut are in different packets? (VL Dolnikov)

There are seven piles with $2014$ pebbles each and a pile with $2008$ pebbles. Ana and Beto play in turns and Ana always plays first. One move consists of removing pebbles from all the piles. From each pile is removed a different amount of pebbles, between $1$ and $8$ pebbles. The first player who cannot make a move loses. a) Who has a winning strategy? b) If there were seven piles with $2015$ pebbles each and a pile with $2008$ pebbles, who has a winning strategy?

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3,\hdots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$.

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

$10$ integers are written in a row. A second row of $10$ integers is formed as follows: the integer written under each integer $A$ of the first row is equal to the total number of integers that stand to the right side of $A$ (in the first row) and are strictly greater than A. A third row is formed by the same way under the second one, and so on. (a) Prove that after several steps a “zero row” (i.e. a row consisting entirely of zeros) appears. (b) What is the maximal possible number of non-zero rows (i.e. rows in which at least one entry is not zero)? (S Tokarev)

Prove that for every convex polygon, we can choose three of its consecutive vertices, such that the circle, defined by them, covers the the entire polygon. (proposed by J. Tabov)

A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations.

a) Let $n$ $(n \geq 2)$ be an integer. On a line there are $n$ distinct (pairwise distinct) sets of points, such that for every integer $k$ $(1 \leq k \leq n)$ the union of every $k$ sets contains exactly $k+1$ points. Show that there is always a point that belongs to every set. b) Is the same conclusion true if there is an infinity of distinct sets of points such that for every positive integer $k$ the union of every $k$ sets contains exactly $k+1$ points?