Found problems: 573
2021 Durer Math Competition Finals, 4
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 $1$.
1997 Denmark MO - Mohr Contest, 1
Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?
1970 All Soviet Union Mathematical Olympiad, 141
All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.
2018 Junior Regional Olympiad - FBH, 3
Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$
2024 China Team Selection Test, 20
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.
1960 Polish MO Finals, 5
From the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ all possible four-digit numbers with different digits are formed. Find the sum of these numbers.
2004 All-Russian Olympiad Regional Round, 10.6
A set of five-digit numbers $\{N_1, ...,N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1$, $...$ , $N_k$. Find the smallest possible value of $k$.
2014 Hanoi Open Mathematics Competitions, 3
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.
VII Soros Olympiad 2000 - 01, 11.4
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$.
Find the first $200$ decimal digits for the number $a^{2000}$.
2013 Costa Rica - Final Round, 6
Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.
1997 Israel Grosman Mathematical Olympiad, 1
Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.
2023 Ukraine National Mathematical Olympiad, 10.1
Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$.
[i]Proposed by Oleksiy Masalitin[/i]
2024 Polish Junior MO Finals, 5
Let $S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}$. Show that the $2S$-digit number
\[\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}\]
is a multiple of $19$.
1976 IMO Longlists, 47
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
1974 Swedish Mathematical Competition, 3
Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.
2015 Estonia Team Selection Test, 7
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.
2021 German National Olympiad, 3
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.
1980 Bundeswettbewerb Mathematik, 1
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?
2011 Denmark MO - Mohr Contest, 5
Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$.
.
2021 Iran MO (2nd Round), 2
Call a positive integer $n$ "Fantastic" if none of its digits are zero and it is possible to remove one of its digits and reach to an integer which is a divisor of $n$ . ( for example , 25 is fantastic , as if we remove digit 2 , resulting number would be 5 which is divisor of 25 ) Prove that the number of Fantastic numbers is finite.
2011 Belarus Team Selection Test, 1
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
2000 Estonia National Olympiad, 2
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 $M$.
1983 All Soviet Union Mathematical Olympiad, 356
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?
2012 Bosnia and Herzegovina Junior BMO TST, 2
Let $\overline{abcd}$ be $4$ digit number, such that we can do transformations on it. If some two neighboring digits are different than $0$, then we can decrease both digits by $1$ (we can transform $9870$ to $8770$ or $9760$). If some two neighboring digits are different than $9$, then we can increase both digits by $1$ (we can transform $9870$ to $9980$ or $9881$). Can we transform number $1220$ to:
$a)$ $2012$
$b)$ $2021$
2015 NZMOC Camp Selection Problems, 1
Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin:
$$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$
Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?