Olympiad problems

2022 Durer Math Competition Finals, 7

The [i]fragments [/i] of a positive integer are the numbers seen when reading one or more of its digits in order. The [i]fragment sum[/i] equals the sum of all the fragments, including the number itself. For example, the fragment sum of $2022$ is $2022+202+022+20+02+22+2+0+2+2 = 2296$. There is another four-digit number with the same fragment sum. What is it? As the example shows, if a fragment occurs multiple times, then all its occurrences are added, and the fragments beginning with $0$ also count (for instance, $022$ is worth $22$).

1964 Bulgaria National Olympiad, Problem 1

Tags: number theory , digit

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 $7$.

1975 All Soviet Union Mathematical Olympiad, 210

Tags: number theory , combinatorics , digit

Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

1978 IMO Shortlist, 3

Tags: modular arithmetic , number theory , digit , decimal representation

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.

1986 Tournament Of Towns, (119) 1

Tags: number theory , digit , 2-digit , sum , difference

We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.

1999 Greece JBMO TST, 3

Tags: number theory , decimal representation , digit

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

2017 IMO Shortlist, N4

Tags: number theory , decimal representation , digit

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?

2013 Hanoi Open Mathematics Competitions, 4

Tags: last , digit , number theory

Let $A$ be an even number but not divisible by $10$. The last two digits of $A^{20}$ are: (A): $46$, (B): $56$, (C): $66$, (D): $76$, (E): None of the above.

1983 IMO Shortlist, 24

Tags: number theory , decimal representation , periodic sequence , non-periodical functions , digit

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$

2024 Germany Team Selection Test, 2

Tags: digit , product of digits , divisibility , number theory

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

1983 All Soviet Union Mathematical Olympiad, 356

Tags: sequence , number theory , periodical , last digit , digit , 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: divide , divisor , number theory , digit

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.

2006 All-Russian Olympiad Regional Round, 8.1

Tags: number theory , digit

Find some nine-digit number $N$, consisting of different digits, such that among all the numbers obtained from $N$ by crossing out seven digits, there would be no more than one prime. Prove that the number found is correct. (If the number obtained by crossing out the digits starts at zero, then the zero is crossed out.)

2018 India PRMO, 25

Tags: number theory , sum , remainder , digit

Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?

1968 German National Olympiad, 4

Tags: geometric sequence , digit , number theory

Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.

2020 AMC 10, 12

Tags: digit

The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point? $\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$

2016 KOSOVO TST, 2

Tags: digit , induction , algebra

Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7

1999 May Olympiad, 1

Tags: number theory , digit

A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits. Find all pairs of consecutive numbers such that both are tricubic.

1996 May Olympiad, 2

Tags: number theory , digit

Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.

2002 Portugal MO, 4

Tags: number theory , divide , digit

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.

1994 Chile National Olympiad, 3

Tags: prime , number theory , digit

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

2015 Caucasus Mathematical Olympiad, 4

Tags: number theory , digit , remainder

Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?

2018 Junior Regional Olympiad - FBH, 3

Tags: digit , 4 digit

Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$

1980 Bundeswettbewerb Mathematik, 1

Tags: game , number theory , divisibility , digit

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?

2010 Saudi Arabia IMO TST, 3

Tags: number theory , digit

Consider the arithmetic sequence $8, 21,34,47,....$ a) Prove that this sequence contains infinitely many integers written only with digit $9$. b) How many such integers less than $2010^{2010}$ are in the sequence?