Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$. a) Would the monster have woken up between the years $2005$ and $2015$? b) Will we be safe in the next $10$ years?

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits. Find all pairs of consecutive numbers such that both are tricubic.

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that $\overline{xyz} = x+y+z+xy+yz+zx+xyz$

$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$

Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$. [i]Proposed by Evan Chang (squareman), USA[/i]

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

Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first. Examples: $f(1992) = 2199$, $f(2000) = 200$. Determine the smallest positive integer $n$ for which $f(n) = 2n$ holds.

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?

Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties: (i) $n$ has exactly $k$ digits (in decimal representation), (ii) all the digits of $n$ are odd, (iii) $n$ is divisible by $5$, (iv) the number $m = n/5$ has $k$ odd digits

All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]