Found problems: 573
A positive integer is [i]happy[/i] if:
1. All its digits are different and not $0$,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there?
Let $N$ be a positive integer such that $N$ and $N^2$ both end in the same four digits $\overline{abcd}$, where $a \ne 0$. What is the four-digit number $\overline{abcd}$?
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 $1$.
Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.
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.
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?
Find all prime numbers of the form $10101...01$.
Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$.
How many times does the digit $1$ occur in its decimal representation?
For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern:
$A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$
$T(n_1)=T(n_2)=...=T(n_{148})$
$T(m_1)=T(m_2)=...=T(m_{148})$
In a three-digit positive integer $M$, the number of hundreds is less than the number of tenths and the number of tenths is less than the number of ones. The arithmetic mean of the integer three-digit numbers obtained by arranging the number $M$ and its numbers ends with the number $5$. Find all such three-digit numbers $M$.
A positive integer $n$ is [i]Mozart[/i] if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times.
Prove that:
1. All Mozart numbers are even.
2. There are infinitely many Mozart numbers.
Let $ a_n$ be the last nonzero digit in the decimal representation of the number $ n!.$ Does the sequence $ a_1, a_2, \ldots, a_n, \ldots$ become periodic after a finite number of terms?
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 a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).
Find all terms of the arithmetic progression $-1, 18, 37, 56, ...$ whose only digit is $5$.
Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.
Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit
number with all digits equal.
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)
[i](Karl Czakler)[/i]
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.
Let $a$ and $b$ be positive integers such that
(i) both $a$ and $b$ have at least two digits;
(ii) $a + b$ is divisible by $10$;
(iii) $a$ can be changed into $b$ by changing its last digit.
Prove that the hundreds digit of the product $ab$ is even.
A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits.
Find all pairs of consecutive numbers such that both are tricubic.
On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?
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 $55$.