Olympiad problems

1996 Tournament Of Towns, (512) 5

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

1957 Moscow Mathematical Olympiad, 350

Tags: Digits , number theory , distance

The distance between towns $A$ and $B$ is $999$ km. At every kilometer of the road that connects $A$ and $B$ a sign shows the distances to $A$ and $B$ as follows: $\fbox{0-999}$ , $\fbox{1-998}$ ,$\fbox{2-997}$ , $ . . . $ , $\fbox{998-1}$ , $\fbox{999-0}$ How many signs are there, with both distances written with the help of only two distinct digits?

1980 Bundeswettbewerb Mathematik, 1

Tags: game , number theory , Divisibility , Digits

Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?

1964 Bulgaria National Olympiad, Problem 1

Tags: number theory , Digits

A $6n$-digit number is divisible by $7$. Prove that if its last digit is moved to the beginning of the number then the new number is also divisible by $7$.

2021 Puerto Rico Team Selection Test, 4

Tags: number theory , Digits , multiple , divisible

How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$ results in a multiple of $30$?

1966 Kurschak Competition, 2

Tags: number theory , Digits

Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.

OIFMAT III 2013, 1

Tags: number theory , Digits , Perfect Square

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

1989 Austrian-Polish Competition, 3

Tags: primes , number theory , Digits

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

Oliforum Contest V 2017, 1

Tags: number theory , Digits , multiple

We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

1976 IMO Longlists, 47

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.

2004 Peru MO (ONEM), 1

Tags: number theory , Digits

Let $a$ be number of $n$ digits ($ n > 1$). A number $b$ of $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values values of $k$.

2017 Hanoi Open Mathematics Competitions, 5

Tags: number theory , maximum , Digits

Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is (A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.

1975 IMO Shortlist, 5

Tags: decimal representation , Digits , series summation , Inequality , algebra , IMO Shortlist

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

2004 Tournament Of Towns, 5

Tags: number theory , Digits , Digit

Two $10$-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of $10$-digit integers with no two integers being neighbours.

VI Soros Olympiad 1999 - 2000 (Russia), 8.3

Tags: number theory , Digits

$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.

2021 Durer Math Competition Finals, 4

Tags: Digits , number theory

What is the number of $4$-digit numbers that contains exactly $3$ different digits that have consecutive value? Such numbers are for instance $5464$ or $2001$. Two digits in base $10$ are consecutive if their difference is $1$.

1973 Chisinau City MO, 64

Tags: algebra , number theory , Digits

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

2017 Ecuador Juniors, 1

Tags: number theory , Digits

An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$. a) Would the monster have woken up between the years $2005$ and $2015$? b) Will we be safe in the next $10$ years?

1980 Bulgaria National Olympiad, Problem 1

Tags: algebra , Digits , number theory

Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.

2013 Flanders Math Olympiad, 1

Tags: number theory , Digits , divisible , Sum

A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.

2018 May Olympiad, 1

Tags: number theory , Digits , Perfect Squares , Perfect Square

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

2007 Dutch Mathematical Olympiad, 3

Tags: number theory , divisible , Digits

Does there exist an integer having the form $444...4443$ (all fours, and ending with a three) that is divisible by $13$? If so, give an integer having that form that is divisible by $13$, if not, prove that such an integer cannot exist.

1999 All-Russian Olympiad Regional Round, 9.1

Tags: number theory , Digits

All natural numbers from $1$ to $N$, $ N \ge 2$ are written out in a certain order in a circle. Moreover, for any pair of neighboring numbers there is at least one digit appearing in the decimal notation of each of them. Find the smallest possible value of $N$.

2014 Israel National Olympiad, 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]

2016 Junior Regional Olympiad - FBH, 1

Tags: Digits , number theory

Find unknown digits $a$ and $b$ such that number $\overline{a783b}$ is divisible with $56$