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

2015 May Olympiad, 4
Tags: number theory , sum of digits , digit
We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.

2007 Junior Balkan Team Selection Tests - Moldova, 1
Tags: number theory , digit , sum , product , sum of digits
The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

1987 IMO Shortlist, 14
Tags: combinatorics , symmetry , combinatorics of words , counting , digit , recurrence relation
How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

2004 Greece JBMO TST, 3
Tags: number theory , digit
If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number?

2004 All-Russian Olympiad Regional Round, 8.7
Tags: number theory , digit
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.

1994 Tournament Of Towns, (440) 6
Tags: number theory , digit
Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$. (AY Belov,)

2019 Saudi Arabia JBMO TST, 3
Tags: number theory , digit
Find all positive integers of form abcd such that $$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$

2019 Polish Junior MO Second Round, 5.
Tags: number theory , digit
The integer $n \geq 1$ does not contain digits: $1,\; 2,\; 9\;$ in its decimal notation. Prove that one of the digits: $1,\; 2,\; 9$ appears at least once in the decimal notation of the number $3n$.

2013 Saudi Arabia BMO TST, 5
Tags: sum of squares , digit , perfect square , number theory
We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

2011 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory , sum , multiple , digit
a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

2018 Hanoi Open Mathematics Competitions, 5
Tags: number theory , digit
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times  a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$

2007 Postal Coaching, 6
Tags: divisible , digit , divide , number theory
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

2015 CHMMC (Fall), 1
Tags: last digit , digit , number theory
Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

2018 Thailand TST, 2
Tags: decimal representation , digit , number theory
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 $S(m)$?

2022 Durer Math Competition Finals, 1
Tags: number theory , digit
How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

1998 Tournament Of Towns, 2
Tags: number theory , sum , product , digit , 4-digit
For every four-digit number, we take the product of its four digits. Then we add all of these products together . What is the result? ( G Galperin)

2020 HK IMO Preliminary Selection Contest, 4
Tags: algebra , digit
In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?

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 $13$?

2020 AMC 10, 12
Tags: digit
The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point? $\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

2006 All-Russian Olympiad Regional Round, 8.1
Tags: digit , number theory
Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

2014 JBMO TST - Macedonia, 3
Tags: digit , divisibility , number theory
Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

2019 Hanoi Open Mathematics Competitions, 2
Tags: digit , last digit , number theory
What is the last digit of $4^{3^{2019}}$? [b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$

2012 Romania National Olympiad, 4
Tags: digit , number theory
[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 $A$.