This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 573

2001 May Olympiad, 1
Tags: number theory , digit
Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?

2019 Denmark MO - Mohr Contest, 1
Tags: number theory , digit
Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?

1951 Polish MO Finals, 2
Tags: number theory , digit , divide
What digits should be placed instead of zeros in the third and fifth places in the number $3000003$ to obtain a number divisible by $13$?

2006 Cuba MO, 8
Tags: last digit , power of 2 , number theory , digit
Prove that for any integer $k$ ($k \ge 2$) there exists a power of $2$ that among its last $k$ digits, the nines constitute no less than half. For example, for $k = 2$ and $k = 3$ we have the powers $2^{12} = ... 96$ and $2^{53} = ... 992$. [hide=original wording] Probar que para cualquier k entero existe una potencia de 2 que entre sus ultimos k dıgitos, los nueves constituyen no menos de la mitad. [/hide]

2023 Kyiv City MO Round 1, Problem 4
Tags: number theory , algebra , digit
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]

1993 Nordic, 3
Tags: system of equations , digit , positive integer , number theory
Find all solutions of the system of equations $\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$ where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.

2017 Auckland Mathematical Olympiad, 3
Tags: number theory , divisible , digit , divide
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

1995 Bundeswettbewerb Mathematik, 4
Tags: multiple , digit , number theory
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.

2013 Chile National Olympiad, 1
Tags: digit , number theory , combinatorics
Find the sum of all $5$-digit positive integers that they have only the digits $1, 2$, and $5$, none repeated more than three consecutive times.

2007 Estonia National Olympiad, 1
Tags: digit , number theory
Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

2018 Malaysia National Olympiad, A2
Tags: digit , number theory
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

1984 Spain Mathematical Olympiad, 2
Tags: number theory , digit
Find the number of five-digit numbers whose square ends in the same five digits in the same order.

2002 Germany Team Selection Test, 3
Tags: number theory , decimal representation , digit , factorial
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 $k$.

2018 Estonia Team Selection Test, 6
Tags: number theory , digit
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.)

1966 Kurschak Competition, 2
Tags: number theory , digit
Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

2011 May Olympiad, 2
Tags: number theory , digit
We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true : $\bullet$ $a\ge b$ $\bullet$ $ab - cd = cd -ba$. For example, $2011$ is pora because $20-11 = 11-02$ Find all the numbers around.

2016 Middle European Mathematical Olympiad, 7
Tags: parity , digit , number theory
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.

2015 Balkan MO Shortlist, N7
Tags: number theory , digit , decimal representation
Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$? (Bulgaria)

2019 Costa Rica - Final Round, 3
Tags: number theory , digit
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$

2012 Romania National Olympiad, 4
Tags: number theory , digit
[i]Reduced name[/i] of a natural number $A$ with $n$ digits ($n \ge 2$) a number of $n-1$ digits obtained by deleting one of the digits of $A$: For example, the [i]reduced names[/i] of $1024$ is $124$, $104$ and $120$. Determine how many seven-digit numbers cannot be written as the sum of one natural numbers $A$ and a [i]reduced name[/i] of $A$.

2013 Tournament of Towns, 2
Tags: digit , combinatorics , number theory
There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?

1971 All Soviet Union Mathematical Olympiad, 144
Tags: number theory , power of 2 , digit
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).

2007 Puerto Rico Team Selection Test, 5
Tags: number theory , digit
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.

2018 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , digit , palindrome
Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values ​​of $N-M$.

1995 Chile National Olympiad, 4
Tags: cube , combinatorics , digit
It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?