Found problems: 546
VMEO III 2006 Shortlist, N4
Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.
2002 Portugal MO, 4
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.
1999 Tournament Of Towns, 5
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)
2016 Hanoi Open Mathematics Competitions, 1
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ :
(A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.
2008 Flanders Math Olympiad, 1
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.
2004 Austria Beginners' Competition, 1
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits.
(Note: Four digits means that the thousand Unit digit must not be $0$.)
1974 All Soviet Union Mathematical Olympiad, 197
Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.
2000 Bundeswettbewerb Mathematik, 1b
Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?
1993 Tournament Of Towns, (369) 1
Find all integers of the form $2^n$ (where $n$ is a natural number) such that after deleting the first digit of its decimal representation we again get a power of $2$.
2019 Paraguay Mathematical Olympiad, 3
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
1978 IMO Shortlist, 3
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1980 IMO Shortlist, 6
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
2003 Cuba MO, 1
Given the following list of numbers:
$$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$
where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?
VII Soros Olympiad 2000 - 01, 11.4
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$.
Find the first $200$ decimal digits for the number $a^{2000}$.
2021 Iran MO (2nd Round), 2
Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
2011 Belarus Team Selection Test, 1
Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$.
I.Kozlov
2009 Switzerland - Final Round, 2
A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that
$$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.
2012 Bundeswettbewerb Mathematik, 1
Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$?
2017 Latvia Baltic Way TST, 7
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?
2018 Saudi Arabia GMO TST, 2
Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.
1951 Polish MO Finals, 2
What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?
2016 India PRMO, 3
Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$.
Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.
2015 Cuba MO, 4
Let $A = \overline{abcd}$ be a $4$-digit positive integer, such that $a\ge 7$ and $a > b >c > d > 0$. Let us consider a positive integer $B = \overline{dcba}$. If all digits of $A+B$ are odd, determine all possible values of $A$.
2000 Belarus Team Selection Test, 6.2
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$.
Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.
2002 Estonia National Olympiad, 2
Do there exist distinct non-zero digits $a, b$ and $c$ such that the two-digit number $\overline{ab}$ is divisible by $c$, the number $\overline{bc}$ is divisible by $a$ and $\overline{ca}$, is divisible by $b$?