Olympiad problems

2011 Tournament of Towns, 4

Positive integers $a < b < c$ are such that $b + a$ is a multiple of $b - a$ and $c + b$ is a multiple of $c-b$. If $a$ is a $2011$-digit number and $b$ is a $2012$-digit number, exactly how many digits does $c$ have?

2021 Israel National Olympiad, P4

Tags: combinatorics , Digits

Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$. Is the amount of numbers Danny likes divisible by $7$?

1975 IMO Shortlist, 6

Tags: modular arithmetic , Divisibility , Digits , decimal representation , IMO , IMO 1975 , digit sum

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

1998 Israel National Olympiad, 2

Tags: decimal representation , Digits , number theory

Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.

1997 Singapore Senior Math Olympiad, 3

Tags: number theory , Digits , Perfect Square

Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

1956 Putnam, A2

Tags: Putnam , number theory , Digits

Prove that every positive integer has a multiple whose decimal representation involves all ten digits.

2017 May Olympiad, 1

Tags: number theory , Digits

We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

2009 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , Digits , multiple

For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times. Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.

2008 Regional Olympiad of Mexico Northeast, 3

Tags: number theory , Digits

Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.

2010 Cuba MO, 1

Tags: number theory , Digits

The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?

2014 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , divides , divisible , Digits

The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2011 Portugal MO, 1

Tags: number theory , Digits

A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?

2023 Chile Junior Math Olympiad, 1

Tags: number theory , Digits

Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.

2019 Portugal MO, 2

Tags: number theory , Digits

A five-digit integer is said to be [i]balanced [/i]i f the sum of any three of its digits is divisible by any of the other two. How many [i]balanced [/i] numbers are there?

2002 Estonia National Olympiad, 3

Tags: number theory , Digits

The teacher writes a $2002$-digit number consisting only of digits $9$ on the blackboard. The first student factors this number as $ab$ with $a > 1$ and $b > 1$ and replaces it on the blackboard by two numbers $a'$ and $b'$ with $|a-a'| = |b-b'| = 2$. The second student chooses one of the numbers on the blackboard, factors it as $cd$ with $c > 1$ and $d > 1$ and replaces the chosen number by two numbers $c'$ and $d'$ with $|c-c'| = |d-d'| = 2$, etc. Is it possible that after a certain number of students have been to the blackboard all numbers written there are equal to $9$?

1971 All Soviet Union Mathematical Olympiad, 144

Tags: number theory , power of 2 , Digits

Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).

2022 Cono Sur, 3

Tags: number theory , Digits , power of number

Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation. For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.

2004 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , Digits

A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.

2019 Durer Math Competition Finals, 12

Tags: number theory , polynomial , last digits , Digits

$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?

2010 CHMMC Fall, 1

Tags: number theory , Digits , CHMMC

The numbers $25$ and $76$ have the property that when squared in base $10$, their squares also end in the same two digits. A positive integer is called [i]amazing [/i] if it has at most $3$ digits when expressed in base $21$ and also has the property that its square expressed in base $21$ ends in the same $3$ digits. (For this problem, the last three digits of a one-digit number b are 00b, and the last three digits of a two-digit number $\underline{ab}$ are $0\underline{ab}$.) Compute the sum of all amazing numbers. Express your answer in base $21$.

1990 IMO Longlists, 65

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.

2014 Contests, 1

Tags: number theory , modular arithmetic , Digits

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

2023 Israel National Olympiad, P7

Tags: number theory , Digits , guessing game

Ana and Banana are playing a game. Initially, Ana secretly picks a number $1\leq A\leq 10^6$. In each subsequent turn of the game, Banana may pick a positive integer $B$, and Ana will reveal to him the most common digit in the product $A\cdot B$ (written in decimal notation). In the case when at least two digits are tied for being the most common, Ana will reveal all of them to Banana. For example, if $A\cdot B=2022$, Ana will tell Banana that the digit $2$ is the most common, while if $A\cdot B=5783783$, Ana will reveal that $3, 7$ and $8$ are the most common. Banana's goal is to determine with certainty the number $A$ after some number of turns. Does he have a winning strategy?

2016 Dutch IMO TST, 3

Tags: number theory , sum of digits , Digits

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 $s(n) > s(2n)$.

1996 All-Russian Olympiad Regional Round, 8.2

Tags: number theory , Digits

Let's call a ticket with a number from $000000$ to $999999$ [i]excellent [/i] if the difference between some two adjacent digits is $5$. Find the number of excellent tickets.