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

1994 Tournament Of Towns, (440) 6
Tags: Digits , number theory
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,)

2013 Thailand Mathematical Olympiad, 5
Tags: sum of digits , number theory , Digits , divides , divisible
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

2005 May Olympiad, 1
Tags: number theory , Digits
Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.

1966 IMO Shortlist, 54
Tags: number theory , decimal representation , Digits , Last digit , modular arithmetic , IMO Shortlist , IMO Longlist
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

2007 Estonia National Olympiad, 1
Tags: number theory , divided , Digits
The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

2013 May Olympiad, 2
Tags: algebra , Digits , number theory
Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.

1962 Poland - Second Round, 6
Tags: number theory , Digits
Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.

2018 Thailand TST, 2
Tags: IMO Shortlist , number theory , decimal representation , Digits , IMO , IMO 2017
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 $S(m)$?

2000 May Olympiad, 1
Tags: number theory , Digits
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.

1987 Tournament Of Towns, (150) 1
Tags: number theory , Digits , last digits , power of 3
Prove that the second last digit of each power of three is even . (V . I . Plachkos)

1989 Tournament Of Towns, (205) 3
Tags: number theory , divisible , divides , Digits
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

2018 Federal Competition For Advanced Students, P1, 3
Tags: combinatorics , game , number theory , decimal representation , Digits
Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

1965 German National Olympiad, 5
Tags: Digits , number theory
Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.

1982 Austrian-Polish Competition, 4
Tags: number theory , Sequence , recurrence relation , bounded , unbounded , Digits , product of digits
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.

1985 All Soviet Union Mathematical Olympiad, 396
Tags: sum of digits , Sum of Squares , Sum , number theory , decimal representation , Digits
Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2015 Cuba MO, 3
Tags: number theory , Digits
Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.

2025 Bundeswettbewerb Mathematik, 2
Tags: number theory , number theory proposed , factorial , Digits
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

2012 Tournament of Towns, 1
Tags: Digits , number theory , power of 2
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?

2014 Hanoi Open Mathematics Competitions, 3
Tags: Sequence , Digits , algebra
How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ? (A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.

2019 Dutch Mathematical Olympiad, 1
Tags: number theory , Digits
A [i]complete [/i] number is a $9$ digit number that contains each of the digits $1$ to $9$ exactly once. The [i]difference [/i] number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the [i]difference [/i] number of $25143$ is equal to $3431$. The [i]complete [/i] number $124356879$ has the additional property that its [i]difference [/i] number, $12121212$, consists of digits alternating between $1$ and $2$. Determine all $a$ with $3 \le a \le 9$ for which there exists a [i]complete [/i] number $N$ with the additional property that the digits of its [i]difference[/i] number alternate between $1 $ and $a$.

2023 Indonesia MO, 4
Tags: number theory , Digits , NT construction , Indonesia , Indonesia MO
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.

2024 Malaysian IMO Training Camp, 2
Tags: number theory , Digits
A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

1985 Swedish Mathematical Competition, 2
Tags: number theory , Digits
Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.

2001 All-Russian Olympiad Regional Round, 9.6
Tags: number theory , last digits , Digits , Divisors
Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

1989 Chile National Olympiad, 1
Tags: number theory , sum of digits , Digits
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$