Given a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Let $a_1$, $a_2$, $a_3$, $...$ denote the strictly increasing sequence of all positive integers $n$ such that $s(7n + 1) = 7s(n) + 1$. Compute $a_{2023}$.

Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy \[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

Let $S_n = n^2 + 20n + 12$, $n$ a positive integer. What is the sum of all possible values of $n$ for which $S_n$ is a perfect square?

Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.

Let $m,n\in\mathbb{Z_{+}}$ be such numbers that set $\{1,2,\ldots,n\}$ contains exactly $m$ different prime numbers. Prove that if we choose any $m+1$ different numbers from $\{1,2,\ldots,n\}$ then we can find number from $m+1$ choosen numbers, which divide product of other $m$ numbers.

Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square. [i]Proposed by Mongolia[/i]

Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities. Remember that the number $1$ is not prime.

Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions.

Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.

$m < n$ are positive integers. Let $p=\frac{n^2+m^2}{\sqrt{n^2-m^2}}$. [b](a)[/b] Find three pairs of positive integers $(m,n)$ that make $p$ prime. [b](b)[/b] If $p$ is prime, then show that $p \equiv 1 \pmod 8$.

Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.

Do there exist (a) four (b) five distinct positive integers such that the sum of any three of them is a prime number? (V Senderov)

Find all pairs of non-negative integers $(x,y)$ such that \[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]

Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$ the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

Suppose that $p|(2t^2-1)$ and $p^2|(2st+1)$. Prove that $p^2|(s^2+t^2-1)$

Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

[b]p1. [/b] If $f(x) = 3x - 1$, what is $f^6(2) = (f \circ f \circ f \circ f \circ f \circ f)(2)$? [b]p2.[/b] A frog starts at the origin of the $(x, y)$ plane and wants to go to $(6, 6)$. It can either jump to the right one unit or jump up one unit. How many ways are there for the frog to jump from the origin to $(6, 6)$ without passing through point $(2, 3)$? [b]p3.[/b] Alfred, Bob, and Carl plan to meet at a café between noon and $2$ pm. Alfred and Bob will arrive at a random time between noon and $2$ pm. They will wait for $20$ minutes or until $2$ pm for all $3$ people to show up after which they will leave. Carl will arrive at the café at noon and leave at $1:30$ pm. What is the probability that all three will meet together? [b]p4.[/b] Let triangle $ABC$ be isosceles with $AB = AC$. Let $BD$ be the altitude from $ B$ to $AC$, $E$ be the midpoint of $AB$, and $AF$ be the altitude from $ A$ to $BC$. If $AF = 8$ and the area of triangle $ACE$ is $ 8$, find the length of $CD$. [b]p5.[/b] Find the sum of the unique prime factors of $(2018^2 - 121) \cdot (2018^2 - 9)$. [b]p6.[/b] Compute the remainder when $3^{102} + 3^{101} + ... + 3^0$ is divided by $101$. [b]p7.[/b] Take regular heptagon $DUKMATH$ with side length $ 3$. Find the value of $$\frac{1}{DK}+\frac{1}{DM}.$$ [b]p8.[/b] RJ’s favorite number is a positive integer less than $1000$. It has final digit of $3$ when written in base $5$ and final digit $4$ when written in base $6$. How many guesses do you need to be certain that you can guess RJ’s favorite number? [b]p9.[/b] Let $f(a, b) = \frac{a^2+b^2}{ab-1}$ , where $a$ and $b$ are positive integers, $ab \ne 1$. Let $x$ be the maximum positive integer value of $f$, and let $y$ be the minimum positive integer value of f. What is $x - y$ ? [b]p10.[/b] Haoyang has a circular cylinder container with height $50$ and radius $5$ that contains $5$ tennis balls, each with outer-radius $5$ and thickness $1$. Since Haoyang is very smart, he figures out that he can fit in more balls if he cuts each of the balls in half, then puts them in the container, so he is ”stacking” the halves. How many balls would he have to cut up to fill up the container? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.

$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.