A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.

Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple $x$, $y$, $z$. Find a triple of positive integers $x$, $y$, $z$ such that $xy + yz + zx = 1000$ (*) and the number of operations that Mazo can subsequently perform with the triple $x, y, z$ is (a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations); (b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).

In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?

Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values ​​of $n$.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

Let's call the number string $D = d_{n-1}d_{n-2}...d_0$ a [i]stable ending[/i] of a number , if for any natural number $m$ that ends in $D$, any of its natural powers $m^k$ also ends in $D$. Prove that for every natural number $n$ there are exactly four stable endings of a number of length $n$. [hide=original wording]Ciparu virkni $D = d_{n-1}d_{n-2}...d_0$ sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe $m^k$ beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.[/hide]

Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?

For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.

Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

How many three-digit positive integers have an odd number of even digits? $\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.

Determine the number of positive integers $n$ satisfying: [list] [*] $n<10^6$ [*] $n$ is divisible by 7 [*] $n$ does not contain any of the digits 2,3,4,5,6,7,8. [/list]

Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^2$ and $b^2$ end in the same last two digits. For example, $35^2 = 1225$ and $45^2 = 2025$ both end in $25$. What are all possible values for the average of $a$ and $b$? For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^2$ to end with the digits 04, not $4$.

Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_ {32} = 6$, $p_ {203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$. Find the largest prime that divides $S $.

How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?

Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.