Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.

Two different prime numbers $p$ and $q$ differ in less than $2$ times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is $p$, and largest prime divisor of second number is $q$.

Suppose that $p$ is a prime number. Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have \[ n|a^{\frac{\varphi(n)}{p}}-1 \]

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

A positive integer $b$ is called good if there exist positive integers $1=a_1, a_2, \ldots, a_{2023}=b$ such that $|a_{i+1}-a_i|=2^i$. Find the number of the good integers.

Determine all values of $n$ satisfied the following condition: there's exist a cyclic $(a_1,a_2,a_3,...,a_n)$ of $(1,2,3,...,n)$ such that $\left\{ {{a_1},{a_1}{a_2},{a_1}{a_2}{a_3},...,{a_1}{a_2}...{a_n}} \right\}$ is a complete residue systems modulo $n$.

There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$. (a) The first machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (a + 13, b + 5)$ If the input pair is $(25,32)$, is it possible to obtain the pair $(82,98)$ after a series of transformations? (b) The second machine can perform two transformations: - $(a, b) \to (a - 1, b - 1)$ - $(a, b) \to (2a, 2b)$ If the input pair is $(34,60)$, is it possible to obtain the pair $(2000, 2008)$ after a series of transformations? (c) The third machine can perform two transformations: - $(a, b) \to (a - 2, b + 2)$ - $(a, b) \to (2a - b + 1, 2b - 1 - a)$ If the input pair is $(145,220)$, is it possible to obtain the pair $(363,498)$ after a series of transformations?

[b]a)[/b] Prove that $ 4040100 $ divides $ 2009\cdot 2011^{2011}+1. $ [i]Gabriel Iorgulescu[/i] [b]b)[/b] Let be three natural numbers $ x,y,z $ with the property that $ (1+\sqrt 2)^x=y^2+2z^2+2yz\sqrt 2. $ Show that $ x $ is even. [i]Marius Cavachi[/i]

Find all $ f:N\rightarrow N$, such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $

If $d$ is a positive integer such that $d \mid 5+2022^{2022}$, show that $d=2x^2+2xy+3y^2$ for some $x, y \in \mathbb{Z}$ iff $d \equiv 3,7 \pmod {20}$.

Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$

Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$. What is the maximal number of positive integers among the $p_i$?

Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.

Suppose that $p,p+d,p+2d,p+3d,p+4d$, and $p+5d$ are six prime numbers, where $p$ and $d$ are positive integers. Show that $d$ must be divisible by $2,3,$ and $5$.

Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$, such that $f(a)+f(b)+ab \mid a^2f(a)+b^2f(b)+f(a)f(b)$ for all positive integers $a,b$.

[b]p1.[/b] Let $D(n)$ denote the number of positive factors of the integer $n$. For example, $D(6) = 4$ , since the factors of $6$ are $1, 2, 3$ , and $6$ . Note that $D(n) = 2$ if and only if $n$ is a prime number. (a) Describe the set of all solutions to the equation $D(n) = 5$ . (b) Describe the set of all solutions to the equation $D(n) = 6$ . (c) Find the smallest $n$ such that $D(n) = 21$ . [b]p2.[/b] At a party with $n$ married couples present (and no one else), various people shook hands with various other people. Assume that no one shook hands with his or her spouse, and no one shook hands with the same person more than once. At the end of the evening Mr. Jones asked everyone else, including his wife, how many hands he or she had shaken. To his surprise, he got a different answer from each person. Determine the number of hands that Mr. Jones shook that evening, (a) if $n = 2$ . (b) if $n = 3$ . (c) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [b]p3.[/b] Let $n$ be a positive integer. A square is divided into triangles in the following way. A line is drawn from one corner of the square to each of $n$ points along each of the opposite two sides, forming $2n + 2$ nonoverlapping triangles, one of which has a vertex at the opposite corner and the other $2n + 1$ of which have a vertex at the original corner. The figure shows the situation for $n = 2$ . Assume that each of the $2n + 1$ triangles with a vertex in the original corner has area $1$. Determine the area of the square, (a) if $n = 1$ . (b) if $n$ is an arbitrary positive integer (the answer may depend on $n$). [img]https://cdn.artofproblemsolving.com/attachments/1/1/62a54011163cc76cc8d74c73ac9f74420e1b37.png[/img] [b]p4.[/b] Arthur and Betty play a game with the following rules. Initially there are one or more piles of stones, each pile containing one or more stones. A legal move consists either of removing one or more stones from one of the piles, or, if there are at least two piles, combining two piles into one (but not removing any stones). Arthur goes first, and play alternates until a player cannot make a legal move; the player who cannot move loses. (a) Determine who will win the game if initially there are two piles, each with one stone, assuming that both players play optimally. (b) Determine who will win the game if initially there are two piles, each with $n$ stones, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . (c) Determine who will win the game if initially there are $n$ piles, each with one stone, assuming that both players play optimally; $n$ is a positive integer, and the answer may depend on $n$ . [b]p5.[/b] Suppose $x$ and $y$ are real numbers such that $0 < x < y$. Define a sequence$ A_0 , A_1 , A_2, A_3, ...$ by-setting $A_0 = x$ , $A_1 = y$ , and then $A_n= |A_{n-1}| - A_{n-2}$ for each $n \ge 2$ (recall that $|A_{n-1}|$ means the absolute value of $A_{n-1}$ ). (a) Find all possible values for $A_6$ in terms of $x$ and $y$ . (b) Find values of $x$ and $y$ so that $A_{1987} = 1987$ and $A_{1988} = -1988$ (simultaneously). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd. (D Fomin, St Petersburg)

Prove that there exists a set $X \subseteq \mathbb{N}$ which contains exactly 2022 elements such that for every distinct $a, b, c \in X$ the following equality: \[ \gcd(a^n+b^n, c) = 1 \] is satisfied for every positive integer $n$.

A positive integer is called [i]good [/i] if it can be written as the sum of two distinct integer squares. A positive integer is called [i]better [/i]if it can be written in at least two was as the sum of two integer squares. A positive integer is called [i]best [/i] if it can be written in at least four ways as the sum of two distinct integer squares. a) Prove that the product of two good numbers is good. b) Prove that $ 5$ is good, $ 2005$ is better, and $ 2005^2$ is best.

Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.