Found problems: 573
2016 Junior Regional Olympiad - FBH, 1
Find unknown digits $a$ and $b$ such that number $\overline{a783b}$ is divisible with $56$
1998 Estonia National Olympiad, 1
Find the last two digits of $11^{1998}$
1998 All-Russian Olympiad Regional Round, 8.1
Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?
2007 May Olympiad, 2
Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.
1999 All-Russian Olympiad Regional Round, 9.1
All natural numbers from $1$ to $N$, $ N \ge 2$ are written out in a certain order in a circle. Moreover, for any pair of neighboring numbers there is at least one digit appearing in the decimal notation of each of them. Find the smallest possible value of $N$.
2019 Gulf Math Olympiad, 2
1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$.
2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$.
3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?
2002 Germany Team Selection Test, 3
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
2011 Tournament of Towns, 4
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?
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 $n$.
1970 IMO Shortlist, 2
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
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 $2$?
1980 IMO Longlists, 6
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
1957 Moscow Mathematical Olympiad, 350
The distance between towns $A$ and $B$ is $999$ km.
At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows:
$\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$
How many signs are there, with both distances written with the help of only two distinct digits?
2017 Latvia Baltic Way TST, 15
Let's call the number string $D = d_{n-1}d_{n-2}...d_0$ a [i]stable ending[/i] of a number , if for any natural number $m$ that ends in $D$, any of its natural powers $m^k$ also ends in $D$. Prove that for every natural number $n$ there are exactly four stable endings of a number of length $n$.
[hide=original wording]Ciparu virkni $D = d_{n-1}d_{n-2}...d_0$ sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe $m^k$ beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.[/hide]
2010 Bundeswettbewerb Mathematik, 1
Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?
2015 Lusophon Mathematical Olympiad, 2
Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation.
That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.
2001 Estonia Team Selection Test, 5
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
2021 Israel National Olympiad, P4
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 $7$?
2009 Postal Coaching, 4
A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.
2023 Kyiv City MO Round 1, Problem 4
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds:
$$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$
For which largest $k$ do there exist $k$-similar numbers?
[i]Proposed by Oleksiy Masalitin[/i]
1996 Estonia National Olympiad, 2
Does there exist a positive integer such that its last digit is nonzero and that it becomes exactly two times bigger when the order of its digits is reversed?
2019 Girls in Mathematics Tournament, 3
We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions:
$\bullet$ All of its digits are $1$ or $2$
$\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct.
For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?
2017 IMO Shortlist, N4
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 $S(m)$?
2014 Singapore Junior Math Olympiad, 2
Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.
2017 Irish Math Olympiad, 1
Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.