Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

We have \( n \) chips that are initially placed on the number line at position 0. On each move, we select a position \( x \in \mathbb{Z} \) where there are at least two chips; we take two of these chips, then place one at \( x-1 \) and the other at \( x+1 \). a) Prove that after a finite number of moves, regardless of how the moves are chosen, we will reach a final position where no two chips occupy the same number on the number line. b) For every possible final position, let \( \Delta \) represent the difference between the numbers where the rightmost and the leftmost chips are located. Find all possible values of \( \Delta \) in terms of \( n \).

$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.

Numbers from $1$ to $1000$ are arranged around a circle. Prove that it is possible to form $500$ non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than $749$. (AA Razborov, Moscow)

Let $S$ be a nonempty set of points of the plane. We say that $S$ determines the distance $d > 0$ if there are two points $A, B$ in $S$ such that $AB = d$. Assuming that $S$ does not contain $8$ collinear points and that it determines not more than $91$ distances, prove that $S$ has less than $2004$ elements.

We say that a digit is [i]high[/i] if it is placed between two other digits and it is bigger than both of them. The digits $0$,$1$,$2$,$\dots$,$9$ are used exactly once to form a 10-digit number. How many numbers can be formed with the property such that they don’t have any high digits?

One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy. Prove that a board consisting of $10$ persons can be elected making every member of the society happy.

[b]p1.[/b] Find the last digit of $2022^{2022}$. [b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$. [b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$. [b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$. [b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$. [b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$. [b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. [b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider permutations $(a_0, a_1, . . . , a_{2022})$ of $(0, 1, . . . , 2022)$ such that $\bullet$ $a_{2022} = 625$, $\bullet$ for each $0 \le i \le 2022$, $a_i \ge \frac{625i}{2022}$ , $\bullet$ for each $0 \le i \le 2022$, $\{a_i, . . . , a_{2022}\}$ is a set of consecutive integers (in some order). The number of such permutations can be written as $\frac{a!}{b!c!}$ for positive integers $a, b, c$, where $b > c$ and $a$ is minimal. Compute $100a + 10b + c$.

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

We color some positive integers $(1,2,...,n)$ with color red, such that any triple of red numbers $(a,b,c)$(not necessarily distincts) if $a(b-c)$ is multiple of $n$ then $b=c$. Prove that the quantity of red numbers is less than or equal to $\varphi(n)$.

Let $A_1$, $A_2$, $\ldots$, $A_n$, $n\geq 2$ be $n$ finite sets with the properties i) $|A_i| \geq 2$, for all $1\leq i \leq n$; ii) $|A_i\cap A_j| \neq 1$, for all $1\leq i<j\leq n$. Prove that the elements of the set $\displaystyle \bigcup_{i=1}^n A_i$ can be colored with 2 colors, such that all the sets $A_i$ are bi-color, for all $1\leq i \leq n$.

Let $n\ge 3$ be an integer. Put $n^2$ cards, each labelled $1,2,\ldots ,n^2$ respectively, in any order into $n$ empty boxes such that there are exactly $n$ cards in each box. One can perform the following operation: one first selects $2$ boxes, takes out any $2$ cards from each of the selected boxes, and then return the cards to the other selected box. Prove that, for any initial order of the $n^2$ cards in the boxes, one can perform the operation finitely many times such that the labelled numbers in each box are consecutive integers.

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails. With how many different directions of impact can the ball be brought onto such a track?

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) 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].

Divide the set of twelve numbers $A = \{3, 4, 5, ...,13, 14\}$ into two sets $ B$ and $C$ 'of six numbers each according to this condition: for any two different numbers with $ B$ their sum does not belong to $ B$ and for any two different numbers from $C$, the sum does not belong to $C$.

On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses $\times$, the other $\circ$. The first who fills a $2\times 2$ square with his symbols wins. Can the player who starts always win?

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

The cells of a table [b]m x n[/b], $m \geq 5$, $n \geq 5$ are colored in 3 colors where: (i) Each cell has an equal number of adjacent (by side) cells from the other two colors; (ii) Each of the cells in the 4 corners of the table doesn’t have an adjacent cell in the same color. Find all possible values for $m$ and $n$.

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]

A $50 \times 50$ square board is tiled by the tetrominoes of the following three types: [img]https://cdn.artofproblemsolving.com/attachments/2/9/62c0bce6356ea3edd8a2ebfe0269559b7527f1.png[/img] Find the greatest and the smallest possible number of $L$ -shaped tetrominoes In the tiling. (Folklore)

Given an integer $n\geq 2,$ colour red exactly $n$ cells of an infinite sheet of grid paper. A rectangular grid array is called special if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly $n$ red cells, let $N$ be the largest number of red cells a special rectangular grid array may contain. Determine the least value $N$ may take over all possible configurations of exactly $n$ red cells