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

2003 May Olympiad, 1
Tags: number theory , Digits , algebra
Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$ Find the sum of all the numbers Pedro wrote.

2014 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , Digits , divisible
Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.

2000 Argentina National Olympiad, 1
Tags: number theory , Digits
The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$. Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.

2018 Pan-African Shortlist, N2
Tags: number theory , Divisibility , Digits
A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?

2018 Malaysia National Olympiad, A2
Tags: Digits , number theory
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

2021 Polish Junior MO Finals, 5
Tags: Digits , Divisibility , number theory
Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.

2016 Israel National Olympiad, 3
Tags: sum of digits , number theory , Digits , modular arithmetic , digit sum
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

2022 OMpD, 3
Tags: combinatorics , Digits
Let $N$ be a positive integer. Initially, a positive integer $A$ is written on the board. At each step, we can perform one of the following two operations with the number written on the board: (i) Add $N$ to the number written on the board and replace that number with the sum obtained; (ii) If the number on the board is greater than $1$ and has at least one digit $1$, then we can remove the digit $1$ from that number, and replace the number initially written with this one (with removal of possible leading zeros) For example, if $N = 63$ and $A = 25$, we can do the following sequence of operations: $$25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5$$ And if $N = 143$ and $A = 2$, we can do the following sequence of operations: $$2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3$$ For what values of $N$ is it always possible, regardless of the initial value of $A$ on the blackboard, to obtain the number $1$ on the blackboard, through a finite number of operations?

2007 Junior Balkan Team Selection Tests - Moldova, 1
Tags: number theory , Digits , Sum , Product , sum of digits
The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

1930 Eotvos Mathematical Competition, 1
Tags: number theory , Digits
How many five-digit multiples of 3 end with the digit 6 ?

1955 Poland - Second Round, 2
Tags: number theory , Digits
Find the natural number $ n $ knowing that the sum $$ 1 + 2 + 3 + \ldots + n$$ is a three-digit number with identical digits.

2018 Singapore Junior Math Olympiad, 1
Tags: multiple , number theory , divides , Digits
Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

1978 IMO, 1
Tags: modular arithmetic , number theory , Digits , decimal representation , IMO , IMO 1978
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.

VMEO IV 2015, 12.4
Tags: binary representation , Sum , number theory , Digits
We call the [i]tribi [/i] of a positive integer $k$ (denoted $T(k)$) the number of all pairs $11$ in the binary representation of $k$. e.g $$T(1)=T(2)=0,\, T(3)=1, \,T(4)=T(5)=0,\,T(6)=1,\,T(7)=2.$$ Calculate $S_n=\sum_{k=1}^{2^n}T(K)$.

1983 All Soviet Union Mathematical Olympiad, 356
Tags: Sequence , number theory , periodical , last digits , Digits , floor function
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

2013 Czech-Polish-Slovak Junior Match, 4
Tags: divides , divisor , number theory , Digits
Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

2022 Austrian MO Regional Competition, 2
Tags: combinatorics , Digits
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]

1972 All Soviet Union Mathematical Olympiad, 168
Tags: Digits , game strategy
A game for two. One gives a digit and the second substitutes it instead of a star in the following difference: $$**** - **** = $$ Then the first gives the next digit, and so on $8$ times. The first wants to obtain the greatest possible difference, the second -- the least. Prove that: 1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour. 2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.

1984 Tournament Of Towns, (056) O4
Tags: number theory , sum of digits , product of digits , Digits
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

2022 Chile National Olympiad, 3
Tags: number theory , prime , Digits
The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?

2007 May Olympiad, 2
Tags: number theory , Digits
Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.

2018 Flanders Math Olympiad, 4
Tags: number theory , Digits
Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

2007 Postal Coaching, 6
Tags: Digits , number theory , divides , divisible
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

2017 Denmark MO - Mohr Contest, 4
Tags: algebra , radical , Digits
Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

2015 India PRMO, 6
Tags: number theory , Digits , sum of digits , Perfect Squares
$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$