Found problems: 546
2016 Dutch IMO TST, 3
Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$.
Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.
1949 Moscow Mathematical Olympiad, 171
* Prove that a number of the form $2^n$ for a positive integer $n$ may begin with any given combination of digits.
1988 Tournament Of Towns, (175) 1
Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?
2017 Romania National Olympiad, 4
Find all prime numbers with $n \ge 3$ digits, having the property:
for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.
2022 IFYM, Sozopol, 4
A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.
2019 Durer Math Competition Finals, 6
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?
2010 Greece JBMO TST, 1
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.
1996 Denmark MO - Mohr Contest, 4
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
2019 Polish Junior MO First Round, 1
The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.
1984 Spain Mathematical Olympiad, 2
Find the number of five-digit numbers whose square ends in the same five digits in the same order.
2012 Romania National Olympiad, 4
[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 $A$.
1995 Bundeswettbewerb Mathematik, 4
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.
II Soros Olympiad 1995 - 96 (Russia), 11.5
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?
1986 Tournament Of Towns, (108) 2
A natural number $N$ is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number $N$ (the digit $0$ is not encountered). What is the maximum number of different digits which there can be in such a representation of $N$?
(S . Fomin, Leningrad)
2010 Princeton University Math Competition, 3
Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.
2023 Brazil National Olympiad, 1
A positive integer is called [i]vaivém[/i] when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, $2021$ is [i]vaivém[/i], as $2 > 0$ and $0 < 2$ and $2 > 1$. The number $2023$ is not [i]vaivém[/i], as $2 > 0$ and $0 < 2$, but $2$ is not greater than $3$.
a) How many [i]vaivém[/i] positive integers are there from $2000$ to $2100$?
b) What is the largest [i]vaivém[/i] number without repeating digits?
c) How many distinct $7$-digit numbers formed by all the digits $1, 2, 3, 4, 5, 6$ and $7$ are [i]vaivém[/i]?
2012 Denmark MO - Mohr Contest, 4
Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.
2024 Czech-Polish-Slovak Junior Match, 5
Is there a positive integer $n$ such that when we write the decimal digits of $2^n$ in opposite order, we get another integer power of $2$?
2024 China Team Selection Test, 20
A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences.
If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.
2018 Auckland Mathematical Olympiad, 1
Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.
2013 Saudi Arabia BMO TST, 5
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 $n$.
OMMC POTM, 2023 7
Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$.
[i]Proposed by Evan Chang (squareman), USA[/i]
1970 All Soviet Union Mathematical Olympiad, 136
Given five $n$-digit binary numbers. For each two numbers their digits coincide exactly on $m$ places. There is no place with the common digit for all the five numbers. Prove that $$2/5 \le m/n \le 3/5$$
1980 IMO Longlists, 6
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
2015 Caucasus Mathematical Olympiad, 4
Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?