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

2018 Estonia Team Selection Test, 6
Tags: number theory , Digits
We call a positive integer $n$ whose all digits are distinct [i]bright[/i], if either $n$ is a one-digit number or there exists a divisor of $n$ which can be obtained by omitting one digit of $n$ and which is bright itself. Find the largest bright positive integer. (We assume that numbers do not start with zero.)

2018 Junior Regional Olympiad - FBH, 3
Tags: Digits , 4 digit
Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$

2009 Tournament Of Towns, 2
Tags: number theory , Digits
Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?

2001 All-Russian Olympiad Regional Round, 9.8
Tags: number theory , Perfect Square , Digits
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.

2002 Germany Team Selection Test, 3
Tags: number theory , decimal representation , Digits , factorial , IMO Shortlist
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

1964 Dutch Mathematical Olympiad, 5
Tags: number theory , product of digits , Product , Digits
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$: (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$.

1976 IMO Shortlist, 11
Tags: number theory , Digits , decimal representation , exponential , IMO Shortlist
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

1992 Czech And Slovak Olympiad IIIA, 3
Tags: sum of digits , Sum , Digits , number theory
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

2001 Portugal MO, 6
Tags: number theory , Digits
Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.

1967 All Soviet Union Mathematical Olympiad, 085
Tags: number theory , Digits
a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

1990 IMO Shortlist, 20
Tags: pigeonhole principle , number theory , decimal representation , Divisibility , multiple , IMO Shortlist , Digits
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.

1990 Greece National Olympiad, 4
Tags: number theory , last digits , Digits
Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

1972 IMO Shortlist, 6
Tags: modular arithmetic , number theory , decimal representation , Digits , IMO Shortlist
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

2022 JBMO Shortlist, N6
Tags: number theory , Sum of Squares , Digits , Junior , Balkan , shortlist
Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.

1994 Tournament Of Towns, (403)
Tags: number theory , Digits
A schoolgirl forgot to write a multiplication sign between two $3$-digit numbers and wrote them as one number. This $6$-digit result proved to be $3$ times greater than the product (obtained by multiplication). Find these numbers. (A Kovaldzhi,

1986 Tournament Of Towns, (119) 1
Tags: number theory , Digits , 2-digit , Sum , Difference
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

1998 May Olympiad, 1
Tags: number theory , Digits
Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

2001 Estonia Team Selection Test, 5
Tags: primes , Product , Power , number theory , Digits
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2022 Kazakhstan National Olympiad, 2
Tags: number theory , Digits , function
We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

1970 IMO Shortlist, 7
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 $a \ ?$

2014 JBMO TST - Macedonia, 3
Tags: number theory , Digits , Divisibility
Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

2021 Auckland Mathematical Olympiad, 3
Tags: number theory , Digits
Alice and Bob are independently trying to figure out a secret password to Cathy’s bitcoin wallet. Both of them have already figured out that: $\bullet$ it is a $4$-digit number whose first digit is $5$. $\bullet$ it is a multiple of $9$; $\bullet$ The larger number is more likely to be a password than a smaller number. Moreover, Alice figured out the second and the third digits of the password and Bob figured out the third and the fourth digits. They told this information to each other but not actual digits. After that the conversation followed: Alice: ”I have no idea what the number is.” Bob: ”I have no idea too.” After that both of them knew which number they should try first. Identify this number

2022 Durer Math Competition Finals, 1
Tags: Digits , number theory
How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

1994 Bundeswettbewerb Mathematik, 1
Tags: number theory , decimal representation , prime numbers , Digits , IMO Shortlist
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2014 Junior Regional Olympiad - FBH, 1
Tags: Digits
If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?