Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then \[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\] For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.

$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).

General Tilly and the Duke of Wallenstein play "Divide and rule!" (Divide et impera!). To this end, they arrange $N$ tin soldiers in $M$ companies and command them by turns. Both of them must give a command and execute it in their turn. Only two commands are possible: The command "[i]Divide![/i]" chooses one company and divides it into two companies, where the commander is free to choose their size, the only condition being that both companies must contain at least one tin soldier. On the other hand, the command "[i]Rule![/i]" removes exactly one tin soldier from each company. The game is lost if in your turn you can't give a command without losing a company. Wallenstein starts to command. a) Can he force Tilly to lose if they start with $7$ companies of $7$ tin soldiers each? b) Who loses if they start with $M \ge 1$ companies consisting of $n_1 \ge 1, n_2 \ge 1, \dotsc, n_M \ge 1$ $(n_1+n_2+\dotsc+n_M=N)$ tin soldiers?

There are $n \ge 2$ positive integers written on the whiteboard. A move consists of three steps: calculate the least common multiple $N$ of all numbers then choose any number $a$ and replace $a$ by $N/a$ . Prove that, using a finite number of moves, you can always make all the numbers on the whiteboard equal to $ 1$.

For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$

Let $a_1,a_2,\cdots$ be a strictly increasing sequence on positive integers. Is it always possible to partition the set of natural numbers $\mathbb{N}$ into infinitely many subsets with infinite cardinality $A_1,A_2,\cdots$, so that for every subset $A_i$, if we denote $b_1<b_2<\cdots$ be the elements of $A_i$, then for every $k\in \mathbb{N}$ and for every $1\le i\le a_k$, it satisfies $b_{i+1}-b_{i}\le k$?

Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?

Pablo and Nacho write together a succession of positive integers of $2006$ terms, according to the following rules: Pablo begins, who in his first turn writes $1$, and from then on, each one in his turn writes an integer positive that is greater than or equal to the last number that the opponent wrote and less than or equal to triple the last number that the opponent wrote. When the two of them have written the $2006$ numbers, the sum $S$ of the first $ 2005$ numbers written (all except the last one) and the sum $T$ of the $2006$ numbers written. If $S$ and $T $ are co-cousins, Nacho wins. Otherwise, Pablo wins. Determine which of the two players has a winning strategy, describe the strategy and demonstrate that it is a winning one.

Let $n>1$ be an integer. A set $S \subset \{ 0,1,2, \ldots, 4n-1\}$ is called [i]rare[/i] if, for any $k\in\{0,1,\ldots,n-1\}$, the following two conditions take place at the same time (1) the set $S\cap \{4k-2,4k-1,4k, 4k+1, 4k+2 \}$ has at most two elements; (2) the set $S\cap \{4k+1,4k+2,4k+3\}$ has at most one element. Prove that the set $\{0,1,2,\ldots,4n-1\}$ has exactly $8 \cdot 7^{n-1}$ rare subsets.

Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column. (Folklore)

Standing in a circle are $99$ girls, each with a candy. In each move, each girl gives her candy to either neighbour. If a girl receives two candies in the same move, she eats one of them. What is the minimum number of moves after which only one candy remains?

Starting with an arbitrary pair (a,b) of vectors on the plane, we are allowed to perform the operations of the following two types: (1) To replace $(a,b)$ with $(a+2kb,b)$ for an arbitrary integer $k \ne 0$; (2) To replace $(a,b)$ with $(a,b+2ka)$ for an arbitrary integer $ k \ne 0$. However, we must change the type of operetion in any step. (a) Is it possible to obtain $((1,0), (2,1))$ from $((1,0), (0,1))$, if the first operation is of the type (1)? (b) Find all pairs of vectors that can be obtained from $((1,0), (0,1))$ (the type of first operation can be selected arbitrarily).

Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.

Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold: $$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$ $$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$ $$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$ $$...$$ $$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$ $$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$ $$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

There is $500$ dollars in a bank. Two bank operations are allowed: to withdraw $300$ dollars from the bank or to deposit $198$ dollars into the bank. These operations can be repeated as many times as necessary but only the money that was initially in the bank can be used. What is the largest amount of money that can be borrowed from the bank? How can this be done? (AK Tolpygo)

Each cell of a $7\times7$ table is painted with one of several colours. It is known that for any two distinct rows the numbers of colours used to paint them are distinct and for any two distinct columns the numbers of colours used to paint them are distinct.What is the maximum possible number of colours in the table?

a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present. b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.

Let $n$ and $a \leq n$ be two positive integers. There's $2n$ people sitting around a circle reqularly. Two people are friend iff one of their distance in the circle is $a$(that is , $a-1$ people are between them). Find all integers $a$ in terms of $n$ st we can choose $n$ of these people , no two of them positioned in front of each other(means they're not antipodes of each other in the circle) and the total friendship between them is an odd number.

China Mathematical Olympiad 2018 Q6 Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I} }x_i\le \sum_{i\in \mathbb{I} }a_i$ , find the maximum value of $x_1x_2\cdots x_n.$

The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.

Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns. Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.

Several tiles congruent to the one shown in the picture below are to be fit inside a $11 \times 11$ square table, with each tile covering $6$ whole unit squares, no sticking out the square and no overlapping. (a) Determine the greatest number of tiles which can be placed this way. (b) Find, with a proof, all unit squares which have to be covered in any tiling with the maximal number of tiles. [img]https://cdn.artofproblemsolving.com/attachments/c/d/23d93e9d05eab94925fc54006fe05123f0dba9.png[/img] Poland

We call a permutation $ \left(a_1, a_2, ..., a_n\right)$ of $ \left(1, 2, ..., n\right)$ [i]quadratic[/i] if there exists at least a perfect square among the numbers $ a_1$, $ a_1 \plus{} a_2$, $ ...$, $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$. Find all natural numbers $ n$ such that all permutations in $ S_n$ are quadratic. [i]Remark.[/i] $ S_{n}$ denotes the $ n$-th symmetric group, the group of permutations on $ n$ elements.

Joao had blue, white and red pearls and with them he made a necklace with $20$ pearls that has as many blue as white pearls. João noticed that, regardless of how he cut the necklace into two parts, both with an even number of pearls, one of the parts would always have more blue pearls than white ones. How many red pearls are in Joao's necklace?