In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: $07344$, $14098$, $27356$, $36429$, $45374$, $52207$, $63822$, $70558$, $85237$, $97665$. What is the combination to the safe?

The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$. a) Prove that the sequence $x_i$ has limit. b) Can this limit be irrational if we have started with the rational number? c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.

Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

Find all four-digit natural numbers formed by two even digits and two odd digits that verify that when multiplied by $2$ four-digit numbers are obtained with all their even digits and when divided by $2$ four-digit natural numbers are obtained with all their odd digits.

Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: [i]the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of $13$, and the second digit of them is the average of the other two.[/i] What are the bus lines that go to Alberto's house?

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]

A positive integer is called [i]vaivém[/i] when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, $2021$ is [i]vaivém[/i], as $2 > 0$ and $0 < 2$ and $2 > 1$. The number $2023$ is not [i]vaivém[/i], as $2 > 0$ and $0 < 2$, but $2$ is not greater than $3$. a) How many [i]vaivém[/i] positive integers are there from $2000$ to $2100$? b) What is the largest [i]vaivém[/i] number without repeating digits? c) How many distinct $7$-digit numbers formed by all the digits $1, 2, 3, 4, 5, 6$ and $7$ are [i]vaivém[/i]?

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.

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 $2$.

We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$) is at position $1$. An integer is called [i]even-steven[/i] if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist). An integer is called [i]oddball[/i] if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist). For example, $3122$ is [i]oddball[/i] but not [i]even-steven[/i], $7$ is both [i]even-steven[/i] and [i]oddball[/i], and $123$ is neither [i]even-steven[/i] nor [i]oddball[/i]. (a) Prove: every oddball integer greater than $9$ can be obtained by adding two [i]oddball [/i] integers. (b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two [i]even-steven[/i] integers.

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

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 $9$)?

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

The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$. Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.

We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)

Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

Let $a$ and $b$ be positive integers, $a < b$, such that in the decimal expansion of the fraction $\dfrac{a}{b} $ the five digits $1,4,2,8,6$ appear somewhere, in that order and consecutively. Determine the lowest possible value $b$ can take .

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?

Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.