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: 546

2007 Puerto Rico Team Selection Test, 5
Tags: number theory , Digits
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.

2020 Regional Competition For Advanced Students, 2
Tags: Digits , number theory
The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)

1989 Tournament Of Towns, (217) 1
Tags: number theory , divides , divisible , Digits
Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

2017 Finnish National High School Mathematics Comp, 3
Tags: number theory , Last digit , Digits
Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

1975 IMO, 4
Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , digit sum
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

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

1988 Tournament Of Towns, (194) 1
Tags: number theory , power of 2 , Digits
Is there a power of $2$ such that it is possible to rearrange the digits, giving another power of $2$?

2008 Hanoi Open Mathematics Competitions, 7
Tags: number theory , 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.

1999 May Olympiad, 1
Tags: number theory , Digits
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.

2016 Puerto Rico Team Selection Test, 2
Tags: number theory , Digits
Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

IV Soros Olympiad 1997 - 98 (Russia), 10.3
Tags: number theory , Arithmetic Progression , Digits
Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

1968 Spain Mathematical Olympiad, 7
Tags: number theory , Digits , power of 2
In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions: a) Can there be only two terms with a certain number of digits? b) Can there be five consecutive terms with the same number of digits? c) Can there be four terms of n digits, followed by four with $n + 1$ digits? d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?

2015 Lusophon Mathematical Olympiad, 2
Tags: number theory , Digit , Digits , decimal representation
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.

1912 Eotvos Mathematical Competition, 1
Tags: number theory , Digits
How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?

2023 Regional Olympiad of Mexico Southeast, 1
Tags: Mexico , Digits
Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

2015 Estonia Team Selection Test, 7
Tags: number theory , Digits , consecutive , Power
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

2017 Hanoi Open Mathematics Competitions, 7
Tags: number theory , Digits
Determine two last digits of number $Q = 2^{2017} + 2017^2$

1996 German National Olympiad, 1
Tags: number theory , Digits
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 $1996n$.

2020 Tournament Of Towns, 1
Tags: Digits , number theory , divisible
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ? Mikhail Evdokimov

2017 Romania National Olympiad, 1
Tags: number theory , Digits
Consider the set $$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$ a) Show that the set $M$ contains no integer. b) Find the smallest and the largest element of $M$

2017 Bosnia and Herzegovina Junior BMO TST, 1
Tags: Digits , number theory
Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that $\overline{xyz} = x+y+z+xy+yz+zx+xyz$

1972 IMO Longlists, 22
Tags: modular arithmetic , number theory , decimal representation , Digits , IMO Shortlist
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

2019 Costa Rica - Final Round, 3
Tags: number theory , Digits
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 $x - y$

2014 Hanoi Open Mathematics Competitions, 3
Tags: number theory , last , Digits
How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? (A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above.

2011 May Olympiad, 2
Tags: Perfect Square , perfect cubes , number theory , Digits
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.