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

2004 Thailand Mathematical Olympiad, 16
Tags: Digits , number theory
What are last three digits of $2^{2^{2004}}$ ?

1999 Singapore MO Open, 2
Tags: number theory , Digits , divisible , divides
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

2015 Junior Regional Olympiad - FBH, 4
Tags: Fraction , Digits
Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

2000 May Olympiad, 3
Tags: Digits , number theory
To write all consecutive natural numbers from $1ab$ to $ab2$ inclusive, $1ab1$ digits have been used. Determine how many more digits are needed to write the natural numbers up to $aab$ inclusive. Give all chances. ($a$ and $b$ represent digits)

2018 Auckland Mathematical Olympiad, 2
Tags: number theory , Digits
Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$. How many times does the digit $1$ occur in its decimal representation?

1992 Bundeswettbewerb Mathematik, 1
Tags: number theory , Digits
Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first. Examples: $f(1992) = 2199$, $f(2000) = 200$. Determine the smallest positive integer $n$ for which $f(n) = 2n$ holds.

2004 Chile National Olympiad, 4
Tags: sum of digits , Digits , number theory
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

2014 Singapore Junior Math Olympiad, 1
Tags: number theory , multiple , Digits
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.

1974 Vietnam National Olympiad, 1
Tags: number theory , Digits , positive integer
Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.

2017 Auckland Mathematical Olympiad, 3
Tags: number theory , divisible , Digits , divides
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 $13$.

1994 Tournament Of Towns, (417) 5
Tags: number theory , Digits , divisible , divides
Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

2001 Mexico National Olympiad, 1
Tags: number theory , Digits
Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.

2017 Macedonia JBMO TST, 5
Tags: number theory , Digits , prime factors , Exponents
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

2019 Bundeswettbewerb Mathematik, 2
Tags: Digits , divisible , Divisibility , number theory
The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

1974 All Soviet Union Mathematical Olympiad, 201
Tags: Digits , number theory , 3-digit
Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.

2013 Israel National Olympiad, 4
Tags: number theory , Digits
Determine the number of positive integers $n$ satisfying: [list] [*] $n<10^6$ [*] $n$ is divisible by 7 [*] $n$ does not contain any of the digits 2,3,4,5,6,7,8. [/list]

2018 India PRMO, 25
Tags: number theory , Sum , remainder , Digits
Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

2018 Federal Competition For Advanced Students, P2, 6
Tags: decimal representation , number theory , Digits
Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

2011 May Olympiad, 2
Tags: number theory , Digits
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.

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

1978 Vietnam National Olympiad, 1
Tags: number theory , Digits , equation
Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.

1991 Greece National Olympiad, 3
Tags: number theory , Digits
Find all 2-digit numbers$ n$ having the property: 'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.

2023 Durer Math Competition Finals, 3
Tags: number theory , Digits
Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

2016 Czech-Polish-Slovak Junior Match, 2
Tags: number theory , divides , Digits
Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits. Czech Republic

2009 Puerto Rico Team Selection Test, 2
Tags: number theory , Digits , Perfect Square , last digits
The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?