Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

Find, with proof, all positive integers $n$ such that neither $n$ nor $n^2$ contain a $1$ when written in base $3$.

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen? (F.L. Nazarov, Leningrad)

Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$?

For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.

Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.

Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?

Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?

We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met: [list] [*]$M$ is a perfect square. [*]The number formed by the first $n$ digits of $M$ is a perfect square. [*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero). [/list] Find a hypersquared number with $2000$ digits.

Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

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

$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?