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

1999 Slovenia National Olympiad, Problem 1
Tags: algebra , Digits
Two three-digit numbers are given. The hundreds digit of each of them is equal to the units digit of the other. Find these numbers if their difference is $297$ and the sum of digits of the smaller number is $23$.

1993 Czech And Slovak Olympiad IIIA, 4
Tags: Sum of powers , Digits , Sum , number theory , Sequence
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?

2018 Ecuador Juniors, 5
Tags: number theory , remainder , Digits
We call a positive integer [i]interesting [/i] if the number and the number with its digits written in reverse order both leave remainder $2$ in division by $4$. a) Determine if $2018$ is an interesting number. b) For every positive integer $n$, find how many interesting $n$-digit numbers there are.

PEN A Problems, 103
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.)

2010 Puerto Rico Team Selection Test, 2
Tags: number theory , Digits
Find two three-digit numbers $x$ and $y$ such that the sum of all other three digit numbers is equal to $600x$.

2001 Estonia National Olympiad, 1
Tags: number theory , Digits , Integer
John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer

2023 Kyiv City MO Round 1, Problem 4
Tags: number theory , algebra , Digits
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds: $$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$ For which largest $k$ do there exist $k$-similar numbers? [i]Proposed by Oleksiy Masalitin[/i]

2004 All-Russian Olympiad Regional Round, 10.6
Tags: number theory , Digits
A set of five-digit numbers $\{N_1, ...,N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1$, $...$ , $N_k$. Find the smallest possible value of $k$.

1998 Tuymaada Olympiad, 6
Tags: number theory , Digits , Digit , periodic , Periodic sequence
Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.

2017 Saudi Arabia BMO TST, 3
Tags: number theory , multiple , Digits
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?

1987 Austrian-Polish Competition, 7
Tags: sum of digits , product of digits , Digits , number theory
For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

2009 Mathcenter Contest, 5
Tags: number theory , Digits , power of 2
For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits. Hint: $\log 2$ is irrational number.

2016 Ecuador NMO (OMEC), 5
Tags: number theory , Digits
Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

2005 Estonia National Olympiad, 5
Tags: number theory , Digits , combinatorics
How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ? (Assume no number starts with zero.)

1969 IMO Shortlist, 40
Tags: combinatorics , decimal representation , counting , Digits , IMO Shortlist , IMO Longlist
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.

1970 IMO Longlists, 59
Tags: modular arithmetic , number theory , Digits , equation , IMO Shortlist
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

2011 Junior Balkan Team Selection Tests - Romania, 3
Tags: number theory , Digits , Sum , multiple
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 $27$

2012 VJIMC, Problem 4
Tags: number theory , Digits
Find all positive integers $n$ for which there exists a positive integer $k$ such that the decimal representation of $n^k$ starts and ends with the same digit.

1995 Chile National Olympiad, 4
Tags: cube , combinatorics , Digits
It is possible to write the numbers $111$, $112$, $121$, $122$, $211$, $212$, $221$ and $222$ at the vertices of a cube, so that the numbers written in adjacent vertices match at most in one digit?

2006 All-Russian Olympiad Regional Round, 8.1
Tags: number theory , Digits
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.)

2016 Regional Olympiad of Mexico Northeast, 6
Tags: number theory , Digits
A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?

2017 Costa Rica - Final Round, N2
Tags: number theory , Digits
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.

2020 Denmark MO - Mohr Contest, 3
Tags: Digits , number theory , divides
Which positive integers satisfy the following three conditions? a) The number consists of at least two digits. b) The last digit is not zero. c) Inserting a zero between the last two digits yields a number divisible by the original number.

2007 Estonia National Olympiad, 1
Tags: Digits , number theory
Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

1996 Argentina National Olympiad, 2
Tags: number theory , Digits
Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.