For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let $f(m)=m+S(m)$. Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$

What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer? $\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]

Determine all integers $1 \le m, 1 \le n \le 2009$, for which \begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}

Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

Prove that, for an arbitrary positive integer $n \in Z_{>0}$, the number $n^2- n + 1$ does not have any prime factors of the form $6k + 5$, for $k \in Z_{>0}$.

$\mathcal{P}(\mathbb{n})$ denotes the collection of all subsets of $\mathbb{N}$. Let $f:\mathbb{N} \longrightarrow \mathcal{P}(\mathbb{n})$ be a function such that $$f(n) = \bigcup_{d|n,d<n,n \geq 2} f(d)$$ Find the number of such functions $f$ for which the range of $f \subseteq$ {$1,2,3....2024$}.

Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.

Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $ \frac{1}{2004}$ is among this summands? Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

Find all positive integers that can be written in the following way $\frac{m^2 + 20mn + n^2}{m^3 + n^3}$ Also, $m,n$ are relatively prime positive integers.

Call a sequence of positive integers $\{a_n\}$ good if for any distinct positive integers $m,n$, one has $$\gcd(m,n) \mid a_m^2 + a_n^2 \text{ and } \gcd(a_m,a_n) \mid m^2 + n^2.$$ Call a positive integer $a$ to be $k$-good if there exists a good sequence such that $a_k = a$. Does there exists a $k$ such that there are exactly $2019$ $k$-good positive integers?

Prove that every positive integer can be represented in the form \[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\] with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.

Find the greatest exponent $k$ for which $2001^{k}$ divides $2000^{2001^{2002}}+2002^{2001^{2000}}$.

Let $n>2$ be a positive integer and let $p$ be a prime. Suppose that the nonzero integers are colored in $n$ colors. Let $a_1,a_2,\ldots,a_{n}$ be integers such that for all $1\le i\le n$, $p^i\nmid a_i$ and $p^{i-1}\mid a_i$. In terms of $n$, $p$, and $\{a_i\}_{i=1}^{n}$, determine if there must exist integers $x_1,x_2,\ldots,x_{n}$ of the same color such that $a_1x_1+a_2x_2+\cdots+a_{n}x_{n}=0$. [i]Ravi Jagadeesan.[/i]

[b]6.1. [/b] Three workers can do some work. Second and the third can together complete it twice as fast as the first, the first and the third can together complete it three times faster than the second. At what time since the first and second can do this job faster than the third? [b]6.2.[/b] Prove that the greatest common divisor of the sum of two numbers and their least common multiple is equal to their greatest common divisor the numbers themselves. [b]6.3.[/b] There were 20 schoolchildren at the consultation and 20 problems were dealt with. It turned out that each student solved two problems and each problem was solved by two schoolchildren. Prove that it is possible to organize the analysis in this way tasks so that everyone solves one problem and all tasks are solved. [hide=original wording] Наконсультациибыло20школьниковиразбиралось20задач. Оказалось, что каждый школьник решил две задачи и каждую задачу решило два школьника. Докажите, что можно так организовать разбор задач, чтобыкаждыйрассказалоднузадачуивсезадачибылирассказаны.[/hide] [b]6.4[/b].Two people Α and Β must get from point Μ to point Ν,located 15 km from M. On foot they can move at a speed of 6 km/h. In addition, they have a bicycle at their disposal, on which υou can drive at a speed of 15 km/h. A and B depart from Μ at the same time, A walks, and B rides a bicycle until meeting pedestrian C, going from N to M. Then B walks and C rides a bicycle to meeting with A, hands him a bicycle, on which he arrives at N. When must pedestrian C leave Nfor A and B to arrive at N simultaneously if he walks at the same speed as A and B? [b]6.5./ 7.1[/b] Prove that out of any six people there will always be three pairs of acquaintances or three pairs of strangers. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

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

Prove that, for any integer $g > 2$, there is a unique three-digit number $\overline{abc}_g$ in base $g$ whose representation in some base $h = g \pm 1$ is $\overline{cba}_h$.

For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

On the blackboard there are written $100$ different positive integers . To each of these numbers is added the $\gcd$ of the $99$ other numbers . In the new $100$ numbers , is it possible for $3$ of them to be equal. [i] (С. Берлов)[/i]

Find all two digit numbers $ \overline{AB}$ such that $ \overline{AB}$ divides $ \overline{A0B}$.

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

[b]p7.[/b] An ant starts at the point $(0, 0)$. It travels along the integer lattice, at each lattice point choosing the positive $x$ or $y$ direction with equal probability. If the ant reaches $(20, 23)$, what is the probability it did not pass through $(20, 20)$? [b]p8.[/b] Let $a_0 = 2023$ and $a_n$ be the sum of all divisors of $a_{n-1}$ for all $n \ge 1$. Compute the sum of the prime numbers that divide $a_3$. [b]p9.[/b] Five circles of radius one are stored in a box of base length five as in the following diagram. How far above the base of the box are the upper circles touching the sides of the box? [img]https://cdn.artofproblemsolving.com/attachments/7/c/c20b5fa21fbd8ce791358fd888ed78fcdb7646.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].