At the junction of some countably infinite number of roads sits a greyhound. On one of the roads a hare runs, away from the junction. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])

In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Santa Claus wants to wrap presents. These are available in $n$ sizes $A_1 <A_2 <...<A_n$, and analogously, there are $n$ packaging sizes $B_1 <B_2 <...<B_n$, where $B_i$ is enough to all gift sizes $A_j$ can be grouped with $j\le i$, but too small for those with $j> i$. On the shelf to the right of Santa Claus are the gifts sorted by size, where the smallest are on the right, of course there can be several gifts of the same size, or none of a size at all. To his left is a shelf with packaging, and also these are sorted from small to large in the same direction. He's brooding in what way he should wrap the gifts and sees two methods for doing this, which depend on his thinking and laziness of movement have been optimized: a) He takes the present closest to him and puts it in the closest packaging, in which it fits in. b) He takes the packaging closest to him and packs in it the closest thing to him gift. In both cases he then does the same again, although of course the one he was using the gift and its packaging are missing, and so on. Once it is not large enough if the packaging or the present is not small enough, he / she will provide the present or the packaging back to its place on the shelf and takes the next-closest. Prove that both methods lead to the same result in the end, they are considered to be exactly the same gifts packed in the same packaging.

Let $a(n)$ be the number of representations of the positive integer $n$ as an ordered sum of $1$'s and $2$'s. Let $b(n)$ be the number of representations of the positive integer $n$ as an ordered sum of integers greater than $1.$ Show that $a(n)=b(n+2)$ for each $n$.

Given are sheets and the numbers $00, 01, \ldots, 99$ are written on them. We must put them in boxes $000, 001, \ldots, 999$ so that the number on the sheet is the number on the box with one digit erased. What is the minimum number of boxes we need in order to put all the sheets?

Integers $n$ and $k$ satisfy $n > 2023k^3$. Kingdom Kitty has $n$ cities, with at most one road between each pair of cities. It is known that the total number of roads in the kingdom is at least $2n^{3/2}$. Prove that we can choose $3k + 1$ cities such that the total number of roads with both ends being a chosen city is at least $4k$.

A deck of $ 32$ cards has $ 2$ different jokers each of which is numbered $ 0$. There are $ 10$ red cards numbered $ 1$ through $ 10$ and similarly for blue and green cards. One chooses a number of cards from the deck. If a card in hand is numbered $ k$, then the value of the card is $ 2^k$, and the value of the hand is sum of the values of the cards in hand. Determine the number of hands having the value $ 2004$.

A closed self-intersecting broken line intersects each of its segments once. Prove that the number of its segments is even.

Let $n$ be a positive integer . If $\sigma$ is a permutation of the first $n$ positive integers , let $S(\sigma)$ be the set of all distinct sums of the form $\sum_{i=k}^{l} \sigma(i)$ where $1 \leq k \leq l \leq n$ . [b](a)[/b] Exhibit a permutation $\sigma$ of the first $n$ positive integers such that $|S(\sigma)|\geq \left \lfloor{\frac{(n+1)^2}{4}}\right \rfloor $. [b](b)[/b] Show that $|S(\sigma)|>\frac{n\sqrt{n}}{4\sqrt{2}}$ for all permutations $\sigma$ of the first $n$ positive integers .

Eight students were asked to solve $8$ problems (the same set of problems for each of the students). (a) Each problem was solved by $5$ students. Prove that one canfind two students so that each of the problems was solved by at least one of them. (b) If each problem was solved by $4$ students, then it is possible that no such pair of students exists. Prove this. (S Tokarev)

Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)? \[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\] \[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\] (2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.

You are given a word consisting of letters $a;b;c$ You can apply 3 operations on this word: [b]1)[/b] You can write any $3$ letter long $\text{subword}$ in reverse. ($\text{xyz}\rightarrow \text{zyx}$) [b]2)[/b] You can add same $2$ letters between any $2$ consecutive letters. ($\text{xyxy}\rightarrow \text{xy}$[b]zz[/b]$\text{xy}$) [b]3)[/b] You can remove any $\text{subword}$ in the given form $\text{xyyx}$ ($\text{x}$[b]yzzy[/b]$\text{xy}\rightarrow\text{xxy}$) Given these $3$ operations, can you go from $\text{abccab}$ to $\text{baccba}$? (Note: A $\text{subword}$ is a string of consecutive letters from the given word)

$20$ football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select $10$ teams, so that no two of them have yet played each other. ( S . A . Genkin)

Given an integer $ k > 1.$ We call a $ k \minus{}$digits decimal integer $ a_{1}a_{2}\cdots a_{k}$ is $ p \minus{}$monotonic, if for each of integers $ i$ satisfying $ 1\le i\le k \minus{} 1,$ when $ a_{i}$ is an odd number, $ a_{i} > a_{i \plus{} 1};$ when $ a_{i}$ is an even number, $ a_{i}<a_{i \plus{} 1}.$ Find the number of $ p \minus{}$monotonic $ k \minus{}$digits integers.

A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $ (A)$ $ 1$ to right and $ 3$ up, and $ (B)$ $ 2$ to the left and $ 4$ down. $ (a)$ The start position of the kangaroo is $ (0,0)$. Show that it can jump to the point $ (19,95)$ and determine the number of jumps needed. $ (b)$ Show that if the start position is $ (1,0)$, then it cannot reach $ (19,95)$. $ (c)$ If the start position is $ (0,0)$, find all points $ (m,n)$ with $ m,n \ge 0$ which the kangaroo can reach.

Given an $a\times b$ chessboard where $a,b\ge3$, alice wants to use only $L$-dominoes (as the figure shows) to cover this chessboard. How many grids, at least, are covered even times? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNi82LzhmZDkwMGQzZjU3M2QxMzk4Y2NjNDg5ZTMwM2ZmYjJiMWU3MmUwLnBuZw==&rn=MjAxOC1DMy5wbmc=[/img]

In a galaxy far, far away there are $225$ inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The [i]influence[/i] of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions? [i]Proposed by Arsenii Nikolaev, Bogdan Rublov[/i]

Consider the set $\Omega$ formed by the first twenty natural numbers, $\Omega = \{1, 2, . . . , 20\}$ . A nonempty subset $A$ of $\Omega$ is said to be [i]sumfree [/i ] if for all pair of elements$ x, y \in A$, the sum $(x + y)$ is not in $A$, ( $x$ can be equal to $y$). Prove that $\Omega$ has at least $2018$ sumfree subsets.

In each square of a $6\times6$ grid there is a lightning bug on or off. One move is to choose three consecutive squares, either horizontal or vertical, and change the lightning bugs in those $3$ squares from off to on or from on to off. Show if at the beginning there is one lighting bug on and the rest of them off, it is not possible to make some moves so that at the end they are all turned off.

To prepare for an Olympiad $20$ students went to a coach. The coach gave them $20$ problems and it turned out that (a) each of the students solved two problems and (b) each problem was solved by twostudents. Prove that it is possible to organize the coaching so that each student would discuss one of the problems that (s)he had solved, and so that all problems would be discussed.

Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].

During a school year 44 competitions were held. Exactly 7 students won in each of the competition. For any two competitions, there exists exactly 1 student who won both competitions. Is it true that there exists a student who won all the competitions?

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.