Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

For every three-digit number, we take the product of its three digits. Then we add all of these products together. What is the result? (G Galperin)

If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$

How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? (A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above.

Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .

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

Call a year [i]ultra-even[/i] if all of its digits are even. Thus $2000,2002,2004,2006$, and $2008$ are all [i]ultra-even[/i] years. They are all $2$ years apart, which is the shortest possible gap. $2009$ is not an [i]ultra-even[/i] year because of the $9$, and $2010$ is not an ultra-even year because of the $1$. (a) In the years between the years $1$ and $10000$, what is the longest possible gap between two [i]ultra-even[/i] years? Give an example of two ultra-even years that far apart with no [i]ultra-even[/i] years between them. Justify your answer. (b) What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two [i]ultra-even[/i] years? Again, give an example, and justify your answer.

We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

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?