Found problems: 546
2010 Flanders Math Olympiad, 1
How many zeros does $101^{100} - 1$ end with?
2019 Saint Petersburg Mathematical Olympiad, 1
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.
2008 Denmark MO - Mohr Contest, 5
For each positive integer $n$, a new number $t_n$ is formed from the numbers $2^n$ and $5^n$ which consists of the digits from $2^n$ followed by the digits from $5^n$. For example, $t_4$ is $16625$. How many digits does the number $t_{2008}$ have?
1978 IMO Longlists, 7
Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.
1970 All Soviet Union Mathematical Olympiad, 141
All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.
2003 Paraguay Mathematical Olympiad, 1
How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?
2016 Middle European Mathematical Olympiad, 7
A positive integer $n$ is [i]Mozart[/i] if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times.
Prove that:
1. All Mozart numbers are even.
2. There are infinitely many Mozart numbers.
2008 Chile National Olympiad, 1
Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: [i]the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of $13$, and the second digit of them is the average of the other two.[/i]
What are the bus lines that go to Alberto's house?
2024 Kyiv City MO Round 2, Problem 1
For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime?
[i]Proposed by Oleksii Masalitin[/i]
1999 Bundeswettbewerb Mathematik, 2
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$.
Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.
2015 Caucasus Mathematical Olympiad, 1
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?
1994 Chile National Olympiad, 3
Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.
2004 May Olympiad, 1
Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.
2004 German National Olympiad, 3
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
2019 IberoAmerican, 1
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
2015 Balkan MO Shortlist, N7
Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$?
(Bulgaria)
2020 Centroamerican and Caribbean Math Olympiad, 1
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.
2010 BAMO, 2
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
1998 Mexico National Olympiad, 1
A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.
2016 Kyiv Mathematical Festival, P5
On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?
2009 Tournament Of Towns, 4
We increased some positive integer by $10\%$ and obtained a positive integer. Is it possible that in doing so we decreased the sum of digits exactly by $10\%$ ?
2017 IMO Shortlist, N4
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 $S(m)$?
2020 Chile National Olympiad, 1
Determine all positive integers $n$ such that the decimal representation of the number $6^n + 1$ has all its digits the same.
1966 IMO Longlists, 54
We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$
VMEO III 2006 Shortlist, N14
For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern:
$A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$
$T(n_1)=T(n_2)=...=T(n_{148})$
$T(m_1)=T(m_2)=...=T(m_{148})$