Alex distributes some cookies into several boxes and records the number of cookies in each box. If the same number appears more than once, it is recorded only once. Serge takes one cookie from each box and puts them on the first plate. Then he takes one cookie from each box that is still non-empty and puts the cookies on the second plate. He continues until all the boxes are empty. Then Serge records the number of cookies on each plate. Again, if the same number appears more than once, it is recorded only once. Prove that Alex's record contains the same number of numbers as Serge's record.

A fixed number of people are dividing an inheritance among themselves. An heir will be called poor if he gets less than $\$99$ and rich if he gets more than $\$10 000$ (some heirs may be neither rich nor poor). The total inheritance and the number of heirs are such that the total income of the rich heirs will be no less than that of the poor ones no matter how the inheritance is divided. Prove that the total income of the rich heirs is no less than $100$ times that of the poor ones. (F Nazarov)

Alice has four boxes, $327$ blue balls, and $2022$ red balls. The blue balls are labeled $1$ to $327$. Alice first puts each of the balls into a box, possibly leaving some boxes empty. Then, a random label between $1$ and $327$ (inclusive) is selected, Alice finds the box the ball with the label is in, and selects a random ball from that box. What is the maximum probability that she selects a red ball?

What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers, $0$ and $1$. Prove that it is possible to obtain : a) $1/5$, b) an arbitrary rational number between $0$ and $1$.

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

On an exam there are 5 questions, each with 4 possible answers. 2000 students went on the exam and each of them chose one answer to each of the questions. Find the least possible value of $n$, for which it is possible for the answers that the students gave to have the following property: From every $n$ students there are 4, among each, every 2 of them have no more than 3 identical answers.

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?

A subset $U \subset R$ is open if for any $x \in U$, there exist real numbers $a, b$ such that $x \in (a, b) \subset U$. Suppose $S \subset R$ has the property that any open set intersecting $(0, 1)$ also intersects $S$. Let $T$ be a countable collection of open sets containing $S$. Prove that the intersection of all of the sets of $T$ is not a countable subset of $R$. (A set $\Gamma$ is countable if there exists a bijective function $f : \Gamma \to Z$.)

Inside a convex $100$-gon are selected $k$ points, $2 \leq k \leq 50$. Show that one can choose $2k$ vertices of the $100$-gon so that the convex $2k$-gon determined by these vertices contains all the selected points.

Each field of the table $(mn + 1) \times (mn + 1)$ contains a real number from the interval $[0, 1]$. The sum the numbers in each square section of the table with dimensions $n x n$ is equal to $n$. Determine how big it can be sum of all numbers in the table.

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 $x$. )

Alireza is currently standing at the point $(0, 0)$ in the $x-y$ plane. At any given time, Alireza can move from the point $(x, y)$ to the point $(x + 1, y)$ or the point $(x, y + 1)$. However, he cannot move to any point of the form $(x, y)$ where $y \equiv 2x\,\, (\mod \,\,5)$. Let $p_k$ be the number of paths Alireza can take starting from the point $(0, 0)$ to the point $(k + 1, 2k + 1)$. Evaluate the sum $$\sum^{\infty}_{k=1} \frac{p_k}{5^k}.$$.

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

For a non-empty set $T$ denote by $p(T)$ the product of all elements of $T$. Does there exist a set $T$ of $2021$ elements such that for any $a\in T$ one has that $P(T)-a$ is an odd integer? Consider two cases: 1) All elements of $T$ are irrational numbers. 2) At least one element of $T$ is a rational number.

Points $ A_1,A_2, ... ,A_n$ inside a circle and points $ B_1,B_2,...,B_n$ on its boundary are positioned so that the segments $ A_1B_1,A_2B_2, ... ,A_nB_n$ do not intersect. A bug can go from point $ A_i$ to $ A_j$ if the segment $ A_iA_j$ does not intersect any segment $ A_kB_k$, $ k \neq i, j$. Prove that the bug can go from any point $ A_p$ to any point $ A_q$ in a finite number of steps.

A bookshelf contains $n$ volumes, labelled $1$ to $n$, in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label $k$, takes it out, and inserts it in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.

In a $999 \times 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1,C_2,C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1,C_3$ white and $C_2$ red. Find the maximum value $T$ can attain. [i]Proposed by Merlijn Staps, The Netherlands[/i]

How many ways can you arrange $3$ Alice’s, $1$ Bob, $3$ Chad’s, and $1$ David in a line if the Alice’s are all indistinguishable, the Chad’s are all indistinguishable, and Bob and David want to be adjacent to each other? (In other words, how many ways can you arrange $3$ A’s, $1$ B, $3$ C’s, and $1$ D in a row where the B and D are adjacent?)

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

Serge and Tanya want to show Masha a magic trick. Serge leaves the room. Masha writes down a sequence $(a_1, a_2, \ldots , a_n)$, where all $a_k$ equal $0$ or $1$. After that Tanya writes down a sequence $(b_1, b_2, \ldots , b_n)$, where all $b_k$ also equal $0$ or $1$. Then Masha either does nothing or says “Mutabor” and replaces both sequences: her own sequence by $(a_n, a_{n-1}, \ldots , a_1)$, and Tanya’s sequence by $(1 - b_n, 1 - b_{n-1}, \ldots , 1 - b_1)$. Masha’s sequence is covered by a napkin, and Serge is invited to the room. Serge should look at Tanya’s sequence and tell the sequence covered by the napkin. For what $n$ Serge and Tanya can prepare and show such a trick? Serge does not have to determine whether the word “Mutabor” has been pronounced.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]