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)

2018 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , divisible , Digit

Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

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.

2024 Singapore Junior Maths Olympiad, Q4

Tags: number theory , Digit

Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit? Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]

2021 German National Olympiad, 3

Tags: algebra , sum of cubes , algebra proposed , combinatorics proposed , Digits , Digit

For a fixed $k$ with $4 \le k \le 9$ consider the set of all positive integers with $k$ decimal digits such that each of the digits from $1$ to $k$ occurs exactly once. Show that it is possible to partition this set into two disjoint subsets such that the sum of the cubes of the numbers in the first set is equal to the sum of the cubes in the second set.

2020 MMATHS, I1

Tags: Digit , number

A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$. [i]Proposed by Levi Iszler[/i]

1940 Moscow Mathematical Olympiad, 059

Tags: Digit , number theory , Integer sequence

Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.

2010 Greece JBMO TST, 1

Tags: number theory , Sum of Squares , algebra , Digits , Digit

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

2019 Polish Junior MO First Round, 1

Tags: number theory , last digits , Digit , Digits

The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.

2000 Estonia National Olympiad, 2

Tags: number theory , Digits , Digit

In a three-digit positive integer $M$, the number of hundreds is less than the number of tenths and the number of tenths is less than the number of ones. The arithmetic mean of the integer three-digit numbers obtained by arranging the number $M$ and its numbers ends with the number $5$. Find all such three-digit numbers $M$.

2015 Lusophon Mathematical Olympiad, 2

Tags: number theory , Digit , Digits , decimal representation

Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation. That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.

2022 AMC 10, 3

Tags: AMC , AMC 10 , 2022 AMC , 2022 AMC 10B , Digit , counting

How many three-digit positive integers have an odd number of even digits? $\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , Digits , Digit , divisible , remainder

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?