Found problems: 573
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 $9$?
In the decimal expansion of $\sqrt{2}=1.4142\dots$, Isabelle finds a sequence of $k$ successive zeroes where $k$ is a positive integer.
Show that the first zero of this sequence can occur no earlier than at the $k$-th position after the decimal point.
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 $22$).
For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively.
1) Find all values of n such that $n = P(n)$:
2) Determine all values of n such that $n = S(n) + P(n)$.
Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.
A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions:
$\bullet$ $\overline{abc}$ does not end in $0$.
$\bullet$ $\overline{abc}$ is a multiple of $36$.
$\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$.
Determine all the Ecuadorian numbers.
Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.
How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.
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 $n \geq n_0, d_{n+T} = d_n.$
Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.
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 $7$.
In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.
How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$?
[i](F. Bakharev)[/i]
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 $a_9$.
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]
1. Find $N$, the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$.
2. Find $M$, the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$.
3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$?
Find all $6$ digit natural numbers, which consist of only the digits $1,2,$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.
Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada?
[i]Proposed by Oleksii Masalitin[/i]
Mazo performs the following operation on triplets of non-negative integers:
If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple $x$, $y$, $z$. Find a triple of positive integers $x$, $y$, $z$ such that $xy + yz + zx = 1000$ (*) and the number of operations that Mazo can subsequently perform with the triple $x, y, z$ is
(a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations);
(b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).
Determine the number of ten-digit positive integers with the following properties:
$\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once.
$\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it.
(Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.)
[i](Karl Czakler)[/i]
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.
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?
Sunny wants to send some secret message to usjl. The secret message is a three digit number, where each digit is one digit from $0$ to $9$ (so $000$ is also possibly the secret message). However, when Sunny sends the message to usjl, at most one digit might be altered. Therefore, Sunny decides to send usjl a longer message so that usjl can decipher the message to get the original secret message Sunny wants to send. Sunny and usjl can communicate the strategy beforehand. Show that sending a $4$-digit message does not suffice. Also show that sending a $6$-digit message suffices. If it is deduced that sending a $c$-digit message suffices for some $c>6$, then partial credits may be awarded.