Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

Find all pairs of integers $ (x,y)$ satisfying $ 1\plus{}1996x\plus{}1998y\equal{}xy.$

Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

Prove that there are no rational numbers $x,y,z$ with $x+y+z=0$ and $x^2+y^2+z^2=100$.

We are given a positive integer $s \ge 2$. For each positive integer $k$, we define its [i]twist[/i] $k’$ as follows: write $k$ as $as+b$, where $a, b$ are non-negative integers and $b < s$, then $k’ = bs+a$. For the positive integer $n$, consider the infinite sequence $d_1, d_2, \dots$ where $d_1=n$ and $d_{i+1}$ is the twist of $d_i$ for each positive integer $i$. Prove that this sequence contains $1$ if and only if the remainder when $n$ is divided by $s^2-1$ is either $1$ or $s$.

Let $A$ consist of $16$ elements of the set $\{1, 2, 3,..., 106\}$, so that the difference of two arbitrary elements in $A$ are different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements of $A$ for which their difference equals to $3$.

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

Find all natural numbers that have exactly six divisors whose sum is $3500$.

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]B1 / G1[/b] Find $20^3 + 2^2 + 3^1$. [b]B2[/b] A piece of string of length $10$ is cut $4$ times into strings of equal length. What is the length of each small piece of string? [b]B3 / G2[/b] What is the smallest perfect square that is also a perfect cube? [b]B4[/b] What is the probability a $5$-sided die with sides labeled from $1$ through $5$ rolls an odd number? [b]B5 / G3[/b] Hanfei spent $14$ dollars on chicken nuggets at McDonalds. $4$ nuggets cost $3$ dollars, $6$ nuggets cost $4$ dollars, and $12$ nuggets cost $9$ dollars. How many chicken nuggets did Hanfei buy? [u]Set 2[/u] [b]B6[/b] What is the probability a randomly chosen positive integer less than or equal to $15$ is prime? [b]B7[/b] Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled $1$ through $6$. What is the probability that the sum is greater than $5$? [b]B8 / G4[/b] What is the radius of a circle with area $4$? [b]B9[/b] What is the maximum number of equilateral triangles on a piece of paper that can share the same corner? [b]B10 / G5[/b] Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in $20$ minutes. Bab can make a pizza in $30$ minutes. If Bob and Bab want to make $50$ pizzas in total, how many hours would that take them? [u]Set 3[/u] [b]B11 / G6[/b] Find the area of an equilateral rectangle with perimeter $20$. [b]B12 / G7[/b] What is the minimum possible number of divisors that the sum of two prime numbers greater than $2$ can have? [b]B13 / G8[/b] Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins? [b]B14 / G9[/b] Aven has $4$ distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table? [b]B15 / G10[/b] Find the largest $7$-digit palindrome that is divisible by $11$. PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

a) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $2^a+2^b+2^c$ it's a perfect square. b) Determine if there are positive integers $a, b$ and $c$, not necessarily distinct, such that $a+b+c=2020$ and $3^a+3^b+3^c$ it's a perfect square.

The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?

Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.

All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.

Let $n$ be a positive integer, and define $f(n)=1!+2!+\ldots+n!$. Find polynomials $P$ and $Q$ such that $$f(n+2)=P(n)f(n+1)+Q(n)f(n)$$for all $n\ge1$.

For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, where $P(n)$ and $Q(n)$ are relatively prime. a) Prove that $P(67)$ is not divisible by 3; b) Find all possible $n$, for which $P(n)$ is divisible by 3.

Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.

Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $. Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .

Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$. Find $b - a$.