Olympiad problems

2019 Girls in Mathematics Tournament, 3

We say that a positive integer N is [i]nice[/i] if it satisfies the following conditions: $\bullet$ All of its digits are $1$ or $2$ $\bullet$ All numbers formed by $3$ consecutive digits of $N$ are distinct. For example, $121222$ is nice, because the $4$ numbers formed by $3$ consecutive digits of $121222$, which are $121,212,122$ and $222$, are distinct. However, $12121$ is not nice. What is the largest quantity possible number of numbers that a nice number can have? What is the greatest nice number there is?

2004 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , digit

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 $k$.

2021 Malaysia IMONST 1, 17

Tags: number theory , digit

Determine the sum of all positive integers $n$ that satisfy the following condition: when $6n + 1$ is written in base $10$, all its digits are equal.

1998 Estonia National Olympiad, 1

Tags: digit , number theory

Find the last two digits of $11^{1998}$

2010 Flanders Math Olympiad, 1

Tags: number theory , last digit , digit

How many zeros does $101^{100} - 1$ end with?

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , digit , sum , multiple

a) Prove that if the sum of the non-zero digits $a_1, a_2, ... , a_n$ is a multiple of $27$, then it is possible to permute these digits in order to obtain an $n$-digit number that is a multiple of $27$. b) Prove that if the non-zero digits $a_1, a_2, ... , a_n$ have the property that every ndigit number obtained by permuting these digits is a multiple of $27$, then the sum of these digits is a multiple of $27$

1940 Moscow Mathematical Olympiad, 059

Tags: number theory , integer sequence , digit

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

2014 Hanoi Open Mathematics Competitions, 3

Tags: number theory , last , digit

How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? (A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above.

2002 Belarusian National Olympiad, 8

Tags: number theory , digit

The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them. For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums. (E. Barabanov)

1989 Tournament Of Towns, (217) 1

Tags: number theory , divide , divisible , digit

Find a pair of $2$ six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers. ( M . N . Gusarov, Leningrad)

2002 Estonia National Olympiad, 3

Tags: number theory , digit

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 $9$?

VI Soros Olympiad 1999 - 2000 (Russia), 8.3

Tags: number theory , digit

$72$ 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.

1997 Israel Grosman Mathematical Olympiad, 1

Tags: number theory , digit , prime

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

2011 Belarus Team Selection Test, 1

Tags: number theory , digit , sum , divisible

Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$. I.Kozlov

2018 India PRMO, 3

Tags: number theory , digit , divisible

Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$.

2017 Israel National Olympiad, 2

Tags: number theory , product , digit , maximize

Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$. [list=a] [*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$. [*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$. [/list]

VMEO III 2006 Shortlist, N4

Tags: divisible , digit , number theory , divide

Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.

2019 Polish Junior MO Second Round, 5.

Tags: number theory , digit

The integer $n \geq 1$ does not contain digits: $1,\; 2,\; 9\;$ in its decimal notation. Prove that one of the digits: $1,\; 2,\; 9$ appears at least once in the decimal notation of the number $3n$.

1970 IMO Longlists, 59

Tags: modular arithmetic , digit , equation , number theory

For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$

1992 IMO Longlists, 54

Tags: number theory , decimal representation , digit

Suppose that $n > m \geq 1$ are integers such that the string of digits $143$ occurs somewhere in the decimal representation of the fraction $\frac{m}{n}$. Prove that $n > 125.$

2013 Saudi Arabia BMO TST, 5

Tags: number theory , sum of squares , digit , perfect square

We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

2024 China Team Selection Test, 20

Tags: number theory , digit , geometric sequence

A positive integer is a good number, if its base $10$ representation can be split into at least $5$ sections, each section with a non-zero digit, and after interpreting each section as a positive integer (omitting leading zero digits), they can be split into two groups, such that each group can be reordered to form a geometric sequence (if a group has $1$ or $2$ numbers, it is also a geometric sequence), for example $20240327$ is a good number, since after splitting it as $2|02|403|2|7$, $2|02|2$ and $403|7$ form two groups of geometric sequences. If $a>1$, $m>2$, $p=1+a+a^2+\dots+a^m$ is a prime, prove that $\frac{10^{p-1}-1}{p}$ is a good number.

1947 Moscow Mathematical Olympiad, 134

Tags: digit , number theory

How many digits are there in the decimal expression of $2^{100}$ ?

2019 Istmo Centroamericano MO, 1

Tags: number theory , remainder , digit

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by $4$.

1974 Czech and Slovak Olympiad III A, 3

Tags: number theory , digit , sum of digits , permutation , equation , diophantine equation

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]