Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

Let $p$ be a prime. Prove that $p^2+p+1$ is never a perfect cube.

You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$? Here $(x, y)$ denotes the greatest common divisor of $x, y$. [i]Proposed by Anton Trygub[/i]

Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.

Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.

A sequence $a_0,a_1,a_2,\ldots ,a_n,\ldots$ of positive integers is constructed as follows: [list][*]if the last digit of $a_n$ is less than or equal to $5$ then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits the process stops.) [*]otherwise $a_{n+1}=9a_n$.[/list] Can one choose $a_0$ so that an infinite sequence is obtained?

$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?

Find all sets of positive integers $A=\big\{ a_1,a_2,...a_{19}\big\}$ which satisfy the following: $1\big) a_1+a_2+...+a_{19}=2017;$ $2\big) S(a_1)=S(a_2)=...=S(a_{19})$ where $S\big(n\big)$ denotes digit sum of number $n$.

Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?

There is some natural number $n>1$ on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$ [i]A. Golovanov[/i]

Find all integer/s $n$ such that $\displaystyle{\frac{5^n-1}{3}}$ is a prime or a perfect square of an integer. [i]Proposed by Prajit Adhikari, Nepal[/i]

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.

Let $p$ be a prime number that is greater than $3$. Show that there exist some integers $a_{1}, a_{2}, \cdots a_{k}$ that satisfy: \[-\frac{p}{2}< a_{1}< a_{2}< \cdots <a_{k}< \frac{p}{2}\] making the product: \[\frac{p-a_{1}}{|a_{1}|}\cdot \frac{p-a_{2}}{|a_{2}|}\cdots \frac{p-a_{k}}{|a_{k}|}\] equals to $3^{m}$ where $m$ is a positive integer.

For each positive integer $n$, let \[a_n = \frac {(n + 9)!}{(n - 1)!}.\] Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

Around a circle, $20$ distinct positive integers are written. Alex divides each number by its neighbor, moving clockwise around the circle, and records the remainders obtained in each case. Teo performs a similar process but moves counterclockwise around the circle and records the remainders he obtains. If Alex finds only two distinct remainders among the $20$ he records, determine the number of distinct remainders Teo will record.

[u]Round 1[/u] [b]p1.[/b] The alphabet of the Aau-Bau language consists of two letters: A and B. Two words have the same meaning if one of them can be constructed from the other by replacing any AA with A, replacing any BB with B, or by replacing any ABA with BAB. For example, the word AABA means the same thing as ABA, and AABA also means the same thing as ABAB. In this language, is it possible to name all seven days of the week? [b]p2.[/b] A museum has a $4\times 4$ grid of rooms. Every two rooms that share a wall are connected by a door. Each room contains some paintings. The total number of paintings along any path of $7$ rooms from the lower left to the upper right room is always the same. Furthermore, the total number of paintings along any path of $7$ rooms from the lower right to the upper left room is always the same. The guide states that the museum has exactly $500$ paintings. Show that the guide is mistaken. [img]https://cdn.artofproblemsolving.com/attachments/7/6/0fd93a0deaa71a5bb1599d2488f8b4eac5d0eb.jpg[/img] [b]p3.[/b] A playground has a swing-set with exactly three swings. When 3rd and 4th graders from Dr. Anna’s math class play during recess, she has a rule that if a $3^{rd}$ grader is in the middle swing there must be $4^{th}$ graders on that person’s left and right. And if there is a $4^{th}$ grader in the middle, there must be $3^{rd}$ graders on that person’s left and right. Dr. Anna calculates that there are $350$ different ways her students can arrange themselves on the three swings with no empty seats. How many students are in her class? [img]https://cdn.artofproblemsolving.com/attachments/5/9/4c402d143646582376d09ebbe54816b8799311.jpg[/img] [b]p4.[/b] The archipelago Artinagos has $19$ islands. Each island has toll bridges to at least $3$ other islands. An unsuspecting driver used a bad mapping app to plan a route from North Noether Island to South Noether Island, which involved crossing $12$ bridges. Show that there must be a route with fewer bridges. [img]https://cdn.artofproblemsolving.com/attachments/e/3/4eea2c16b201ff2ac732788fe9b78025004853.jpg[/img] [b]p5.[/b] Is it possible to place the numbers from $1$ to $121$ in an $11\times 11$ table so that numbers that differ by $1$ are in horizontally or vertically adjacent cells and all the perfect squares $(1, 4, 9, ... , 121)$ are in one column? [u]Round 2[/u] [b]p6.[/b] Hungry and Sneaky have opened a rectangular box of chocolates and are going to take turns eating them. The chocolates are arranged in a $2m \times 2n$ grid. Hungry can take any two chocolates that are side-by-side, but Sneaky can take only one at a time. If there are no more chocolates located side-by-side, all remaining chocolates go to Sneaky. Hungry goes first. Each player wants to eat as many chocolates as possible. What is the maximum number of chocolates Sneaky can get, no matter how Hungry picks his? [img]https://cdn.artofproblemsolving.com/attachments/b/4/26d7156ca6248385cb46c6e8054773592b24a3.jpg[/img] [b]p7.[/b] There is a thief hiding in the sultan’s palace. The palace contains $2019$ rooms connected by doors. One can walk from any room to any other room, possibly through other rooms, and there is only one way to do this. That is, one cannot walk in a loop in the palace. To catch the thief, a guard must be in the same room as the thief at the same time. Prove that $11$ guards can always find and catch the thief, no matter how the thief moves around during the search. [img]https://cdn.artofproblemsolving.com/attachments/a/b/9728ac271e84c4954935553c4d58b3ff4b194d.jpg[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for that for every positive integer $ n$, the smallest integer that is greater than $ (\sqrt {3} \plus{} 1)^{2n}$ is divisible by $ 2^{n \plus{} 1}$.

Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]