Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.

Determine all positive integers $n$ such that the decimal representation of the number $6^n + 1$ has all its digits the same.

Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$. $\textit{Proposed by Pablo Serrano, Ecuador}$

Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

A set of five-digit numbers $\{N_1, ...,N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1$, $...$ , $N_k$. Find the smallest possible value of $k$.

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

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.

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 five-digit multiples of 3 end with the digit 6 ?

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$? Note that the first digit $A$ of a four-digit number cannot be zero.

Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$.

Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.

Let $6$ pairwise different digits are given and all of them are different from $0$. Prove that there exist $2$ six-digit integers, such that their difference is equal to $9$ and each of them contains all given $6$ digits.

Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$. Is the amount of numbers Danny likes divisible by $7$?

Consider the natural numbers written in the decimal system. (a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have? (b) Prove that there is no number that decreases $12$ times when its first digit is erased. (c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.