Found problems: 546
2016 Brazil Team Selection Test, 1
For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$
1970 All Soviet Union Mathematical Olympiad, 132
The digits of the $17$-digit number are rearranged in the reverse order. Prove that at list one digit of the sum of the new and the initial number is even.
1986 All Soviet Union Mathematical Olympiad, 430
The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$
2006 Switzerland - Final Round, 3
Calculate the sum of digit of the number
$$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$
where the number of nines doubles in each factor.
2018 Israel National Olympiad, 4
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
2011 Regional Olympiad of Mexico Center Zone, 4
Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.
2018 Regional Olympiad of Mexico Center Zone, 1
Let $M$ and $N$ be two positive five-digit palindrome integers, such that $M <N$ and there is no other palindrome number between them. Determine the possible values of $N-M$.
1984 All Soviet Union Mathematical Olympiad, 386
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
2010 Contests, 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.
2017 Irish Math Olympiad, 1
Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.
1985 Tournament Of Towns, (089) 5
The digits $0, 1 , 2, ..., 9$ are written in a $10 x 10$ table , each number appearing $10$ times .
(a) Is it possible to write them in such a way that in any row or column there would be not more than $4$ different digits?
(b) Prove that there must be a row or column containing more than $3$ different digits .
{ L . D . Kurlyandchik , Leningrad)
1997 Moldova Team Selection Test, 3
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
2002 May Olympiad, 4
In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: $07344$, $14098$, $27356$, $36429$, $45374$, $52207$, $63822$, $70558$, $85237$, $97665$. What is the combination to the safe?
2009 Argentina National Olympiad, 6
A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values of $n$.
Kvant 2019, M2543
Let $a$ and $b$ be 2019-digit numbers. Exactly 12 digits of $a$ are non-zero: the five leftmost and seven rightmost, and exactly 14 digits of $b$ are non-zero: the five leftmost and nine rightmost. Prove that the largest common divisor of $a$ and $b$ has no more than 14 digits.
[i]Proposed by L. Samoilov[/i]
2024 Irish Math Olympiad, P4
How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$?
Note that the first digit $A$ of a four-digit number cannot be zero.
2011 Junior Balkan Team Selection Tests - Romania, 1
Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.
2021 Malaysia IMONST 1, 7
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?
2018 Junior Regional Olympiad - FBH, 4
Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$
2003 Paraguay Mathematical Olympiad, 2
With three different digits, all greater than $0$, six different three-digit numbers are formed. If we add these six numbers together the result is $4.218$. The sum of the three largest numbers minus the sum of the three smallest numbers equals $792$. Find the three digits.
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$
2018 Malaysia National Olympiad, B2
Let $a$ and $b$ be positive integers such that
(i) both $a$ and $b$ have at least two digits;
(ii) $a + b$ is divisible by $10$;
(iii) $a$ can be changed into $b$ by changing its last digit.
Prove that the hundreds digit of the product $ab$ is even.
2019 Durer Math Competition Finals, 14
Let $S$ be the set of all positive integers less than $10,000$ whose last four digits in base $2$ are the same as its last four digits in base $5$. What remainder do we get if we divide the sum of all elements of $S$ by $10000$?
2005 iTest, 31
Let $X = 123456789$. Find the sum of the tens digits of all integral multiples of $11$ that can be obtained by interchanging two digits of $X$.
1991 Greece Junior Math Olympiad, 3
Find the sum of all $4$-digit numbers using the digits $2,3,4,5,6$ without a repetition of any of those digits.