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.

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Found problems: 573

2010 Cuba MO, 1
Tags: number theory , digit
The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?

2000 Estonia National Olympiad, 2
Tags: digit , number theory
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 $M$.

2021 Iran MO (2nd Round), 2
Tags: number theory , digit
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.

1970 IMO Shortlist, 2
Tags: inequalities , algebra , decimal representation , digit
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 $A'B<AB'$.

2000 May Olympiad, 1
Tags: number theory , digit
Find all four-digit natural numbers formed by two even digits and two odd digits that verify that when multiplied by $2$ four-digit numbers are obtained with all their even digits and when divided by $2$ four-digit natural numbers are obtained with all their odd digits.

2016 May Olympiad, 1
Tags: number theory , digit
We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

1965 All Russian Mathematical Olympiad, 059
Tags: number theory , digit
A bus ticket is considered to be lucky if the sum of the first three digits equals to the sum of the last three ($6$ digits in Russian buses). Prove that the sum of all the lucky numbers is divisible by $13$.

2019 Gulf Math Olympiad, 2
Tags: number theory , digit , multiple
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 $2$?

2015 Cuba MO, 4
Tags: number theory , digit
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 $A$.

2020 AMC 10, 19
Tags: digit
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? $\textbf{(A) } 2 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 6 \qquad\textbf{(E) } 7$

2000 Singapore MO Open, 3
Tags: number theory , digit
Is there a positive integer with at most four digits whose value is increased by exactly $60\%$ when the first digit is moved to the end of the number? For example, when the first digit of $1234$ is moved to the end of the number, the result is the integer $2341$.

1992 Czech And Slovak Olympiad IIIA, 3
Tags: sum of digits , sum , digit , number theory
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

2016 Kyiv Mathematical Festival, P5
Tags: combinatorics , digit , divisibility
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?

2007 Regional Olympiad of Mexico Center Zone, 6
Tags: number theory , digit
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values ​​of $N $.

1986 China Team Selection Test, 3
Tags: algebra , number theoretic functions , decimal representation , digit
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 $19^{86}.$

1996 German National Olympiad, 1
Tags: number theory , digit
Find all natural numbers $n$ with the following property: Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.

2021 Middle European Mathematical Olympiad, 8
Tags: perfect square , number theory , digit
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.

2012 Singapore Junior Math Olympiad, 2
Tags: digit , number theory
Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?

2011 Tournament of Towns, 5
Tags: number theory , digit
We will call a positive integer [i]good [/i] if all its digits are nonzero. A good integer will be called [i]special [/i] if it has at least $k$ digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest $k$ such that any good integer can be turned into any other good integer by such moves?

1955 Kurschak Competition, 2
Tags: number theory , divide , digit
How many five digit numbers are divisible by $3$ and contain the digit $6$?

1984 All Soviet Union Mathematical Olympiad, 386
Tags: prime , digit , 3-digit , number theory , prime number
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.

1980 Bulgaria National Olympiad, Problem 1
Tags: algebra , digit , number theory
Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.

2016 CentroAmerican, 1
Tags: number theory , digit
Find all positive integers $n$ that have 4 digits, all of them perfect squares, and such that $n$ is divisible by 2, 3, 5 and 7.

2015 Saudi Arabia JBMO TST, 2
Tags: combinatorics , digit , odd
Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.

2016 Junior Regional Olympiad - FBH, 1
Tags: digit , number theory
Find unknown digits $a$ and $b$ such that number $\overline{a783b}$ is divisible with $56$