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: 573

2022 Austrian MO Regional Competition, 2
Tags: combinatorics , digit
Determine the number of ten-digit positive integers with the following properties: $\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once. $\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it. (Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.) [i](Karl Czakler)[/i]

1968 IMO Shortlist, 22
Tags: number theory , decimal representation , digit , equation
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

1998 Israel National Olympiad, 2
Tags: decimal representation , digit , number theory
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.

2001 Switzerland Team Selection Test, 5
Tags: digit , sum , decimal representation , inequalities , number theory
Let $a_1 < a_2 < ... < a_n$ be a sequence of natural numbers such that for $i < j$ the decimal representation of $a_i$ does not occur as the leftmost digits of the decimal representation of $a_j$ . (For example, $137$ and $13729$ cannot both occur in the sequence.) Prove that $\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19$ .

2014 May Olympiad, 1
Tags: number theory , digit
A natural number $N$ is [i]good [/i] if its digits are $1, 2$, or $3$ and all $2$-digit numbers are made up of digits located in consecutive positions of $N$ are distinct numbers. Is there a good number of $10$ digits? Of $11$ digits?

2022 Durer Math Competition Finals, 1
Tags: digit , 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?

2019 Hanoi Open Mathematics Competitions, 2
Tags: number theory , last digit , digit
What is the last digit of $4^{3^{2019}}$? [b]A.[/b] $0$ [b]B.[/b] $2$ [b]C.[/b] $4$ [b]D.[/b] $6$ [b]E.[/b] $8$

1964 German National Olympiad, 4
Tags: number theory , last digit , periodic , digit
Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!

2020 HK IMO Preliminary Selection Contest, 4
Tags: digit , algebra
In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be chosen in order to maximise the score?

2005 Singapore Senior Math Olympiad, 1
Tags: perfect square , digit , difference , number theory
The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

1968 IMO, 2
Tags: number theory , decimal representation , digit , equation
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.

2018 Federal Competition For Advanced Students, P2, 6
Tags: decimal representation , number theory , digit
Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$. [i](Proposed by Walther Janous)[/i]

1991 Swedish Mathematical Competition, 5
Tags: digit , number theory
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.

2022 JBMO Shortlist, N6
Tags: number theory , sum of squares , digit , 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 $n$.

IV Soros Olympiad 1997 - 98 (Russia), 10.3
Tags: number theory , arithmetic progression , digit
Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

1990 IMO Longlists, 23
Tags: number theory , digit , sum of digits , calculate , decimal representation
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

2015 Estonia Team Selection Test, 7
Tags: number theory , digit , consecutive , power
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

2019 Durer Math Competition Finals, 14
Tags: number theory , last digit , digit
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 $10000$?

2002 Germany Team Selection Test, 3
Tags: number theory , decimal representation , digit , factorial
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 $k$.

1994 All-Russian Olympiad Regional Round, 9.3
Tags: number theory , digit , trinomial
Does there exist a quadratic trinomial $p(x)$ with integer coefficients such that, for every natural number $n$ whose decimal representation consists of digits $1$, $p(n)$ also consists only of digits $1$?

2014 Junior Regional Olympiad - FBH, 1
Tags: digit
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)?

2020 Denmark MO - Mohr Contest, 3
Tags: digit , number theory , divide
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.

2015 Hanoi Open Mathematics Competitions, 2
Tags: digit , number theory
The last digit of number $2017^{2017} - 2013^{2015}$ is (A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.

2005 iTest, 31
Tags: number theory , digit
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 $X$.

2014 Junior Regional Olympiad - FBH, 5
Tags: digit
From digits $0$, $1$, $3$, $4$, $7$ and $9$ were written $5$ digit numbers which all digits are different. How many numbers from them are divisible with $5$