Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

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 .

A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?

All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of$ k$ for which the same $k$-digit number can be found in at least two different places in this string?

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

Sunny wants to send some secret message to usjl. The secret message is a three digit number, where each digit is one digit from $0$ to $9$ (so $000$ is also possibly the secret message). However, when Sunny sends the message to usjl, at most one digit might be altered. Therefore, Sunny decides to send usjl a longer message so that usjl can decipher the message to get the original secret message Sunny wants to send. Sunny and usjl can communicate the strategy beforehand. Show that sending a $4$-digit message does not suffice. Also show that sending a $6$-digit message suffices. If it is deduced that sending a $c$-digit message suffices for some $c>6$, then partial credits may be awarded.

Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

We call a positive integer a [i]shuffle[/i] number if the following hold: (1) All digits are nonzero. (2) The number is divisible by $11$. (3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$. How many $10$-digit [i]shuffle[/i] numbers are there?

Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?

What is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only?

Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.

To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?

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.

The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$

Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation. For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

We have an integer $A$ such that $A^2$ is a four digit number, with $5$ in the ten's place . Find all possible values of $A$.

Find a number that increases by a factor of $1996$ if the digits in the first and fifth places after the decimal place are swapped in its decimal notation.

Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

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

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.

Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?