Consider the following sequence of positive real numbers $\dots<a_{-2}<a_{-1}<a_0<a_1<a_2<\dots$ infinite in both directions. For each positive integer $k$ let $b_k$ be the least integer such that the ratio between the sum of $k$ consecutive terms and the greatest of these $k$ terms is less than or equal to $b_k$(This fact occurs for any sequence of $k$ consecutive numbers). Prove that the sequence $b_1,b_2,b_3,...$ coincides with the sequence $1,2,3,...$ or is eventually constant.

Let \[P(m)=\frac{m}{2} + \frac{m^2}{4}+ \frac{m^4}{8} + \frac{m^8}{8}.\] How many of the values of $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $

Find all prime numbers $p$ and $q$ such that $p^q + 5q - 2$ is also a prime number.

Can two perfect cubes fit between two consecutive perfect squares? In other words, do there exist positive integers $a$, $b$, $n$ such that $n^2 < a^3 < b^3 < (n + 1)^2$? [i](3 points)[/i]

If $\sqrt{\log_bn}=\log_b\sqrt n$ and $b\log_bn=\log_bbn,$ then the value of $n$ is equal to $\frac jk,$ where $j$ and $k$ are relatively prime. What is $j+k$?

How many of the first $1000$ positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?

Prove that for any natural number $ n $ there are $ n $ consecutive numbers, each one of these numbers having the following property: the sum of the positive divisors of a number $ x $ is greater than $ 2x. $

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.

$N$ numbers - ones and twos - are arranged in a circle. We mean a number formed by several digits arranged in a row (clockwise or counterclockwise). For what is the smallest value of $N$, all four-digit numbers whose writing contains only numbers $1$ and $2$, could they be among those shown?

Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers.

[b]p1.[/b] Four positive numbers are placed at the vertices of a rectangle. Each number is at least as large as the average of the two numbers at the adjacent vertices. Prove that all four numbers are equal. [b]p2.[/b] The sum $498+499+500+501=1998$ is one way of expressing $1998$ as a sum of consecutive positive integers. Find all ways of expressing $1998$ as a sum of two or more consecutive positive integers. Prove your list is complete. [b]p3.[/b] An infinite strip (two parallel lines and the region between them) has a width of $1$ inch. What is the largest value of $A$ such that every triangle with area $A$ square inches can be placed on this strip? Justify your answer. [b]p4.[/b] A plane divides space into two regions. Two planes that intersect in a line divide space into four regions. Now suppose that twelve planes are given in space so that a) every two of them intersect in a line, b) every three of them intersect in a point, and c) no four of them have a common point. Into how many regions is space divided? Justify your answer. [b]p5.[/b] Five robbers have stolen $1998$ identical gold coins. They agree to the following: The youngest robber proposes a division of the loot. All robbers, including the proposer, vote on the proposal. If at least half the robbers vote yes, then that proposal is accepted. If not, the proposer is sent away with no loot and the next youngest robber makes a new proposal to be voted on by the four remaining robbers, with the same rules as above. This continues until a proposed division is accepted by at least half the remaining robbers. Each robber guards his best interests: He will vote for a proposal if and only if it will give him more coins than he will acquire by rejecting it, and the proposer will keep as many coins for himself as he can. How will the coins be distributed? Explain your reasoning. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

Dima has red and blue felt—tip pens, with one of them he paints rational points on the numerical axis, and with the other - irrational ones. Dima colored $100$ rational and $100$ irrational points, after which he erased the signatures that allowed to find out where the origin was and what the scale was. Sergey has a compass with which he can measure the distance between any two colored points $A$ and $B$, and then mark on the axis a point located at a measured distance from any colored point $C$ (left or right); at the same time, Dima immediately paints it with the appropriate felt-tip pen. How Sergei can find out what color Dima paints rational points and what color he paints irrational ones?

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Let $a, b$ be coprime positive integers. A positive integer $n$ is said to be [i]weak[/i] if there do not exist any nonnegative integers $x, y$ such that $ax+by=n$. Prove that if $n$ is a [i]weak[/i] integer and $n < \frac{ab}{6}$, then there exists an integer $k \geq 2$ such that $kn$ is [i]weak[/i].

Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$. Show that for all positive integers $r$, \[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]

The set $N$ has been partitioned into two sets A and $B$. Show that for every $n \in N$ there exist distinct integers $a, b > n$ such that $a, b, a + b$ either all belong to $A$ or all belong to $B$.

Suppose $a,b,c$ are positive integers such that \[\gcd(a,b)+\gcd(a,c)+\gcd(b,c)=b+c+2023\] Prove that $\gcd(b,c)=2023$. [i]Remark.[/i] For positive integers $x$ and $y$, $\gcd(x,y)$ denotes their greatest common divisor. [i]Ivan Novak[/i]

Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .

Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$. Here $(a, b)$ denotes the greatest common divisor of integers $a, b$. [i](Proposed by Mykhailo Shtandenko)[/i]

For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6, 8^8,$ and $k$?

Let $f(n)$ be the probability that, if $k \in \{1, 2, ... , 2n\}$ is randomly selected, then $1 + 2 + ... +k$ will be divisible by $n$. Prove that $f(n)$ is distinct for every positive integer $n$.