The distance between towns $A$ and $B$ is $999$ km. At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows: $\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$ How many signs are there, with both distances written with the help of only two distinct digits?

Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

Let $n = 2^{p-1} (2^p - 1)$, and let $2^p- 1$ be a prime number. Prove that the sum of all (positive) divisors of $n$ (not including $n$ itself) is exactly $n$.

Prove that for every natural number $k$ there exists an infinite set of such natural numbers $t$, that the decimal notation of $t$ does not contain zeroes and the sums of the digits of the numbers $t$ and $kt$ are equal.

Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.

We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

The decimal representation of all integers from $1$ to an arbitrary integer $n$ are written one after another as such: $$123... 91011... 99100... (n).$$ Does there exist $n$ such that each of the digits $0,1,2,...,9$ appears the same number of times in the given sequence? (A Andzans)

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits.

Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

The cells of a $2021\times 2021$ table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate $(1, 1)$, contains the number $0$. For every other cell $C$, we consider a route from $(1, 1)$ to $C$, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell $C$. For example, the cell with coordinate $(2, 1)$ contains $1 = 1 + 0$, the cell with coordinate $(3, 1)$ contains $3 = 2 + 0 + 1$, and the cell with coordinate $(3, 2)$ contains $7 = 3 + 0 + 1 + 3$. What is the last digit of the number in the cell $(2021, 2021)$?