Found problems: 573
Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$.
Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.
An eight-digit number is said to be 'robust' if it meets both of the following conditions:
(i) None of its digits is $0$.
(ii) The difference between two consecutive digits is $4$ or $5$.
Answer the following questions:
(a) How many are robust numbers?
(b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all
the super robust numbers.
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find
\[
s_1 + s_2 + \cdots + s_n
\]
A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits.
For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$.
Determine all common sequences.
[i]Proposed by Wong Jer Ren[/i]
Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.
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]
Find the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits.
Czech Republic
How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one?
[i]Proposed by Germany, FR.[/i]
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.
Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?
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 $4$.
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.
Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?
To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits.
(Note: Four digits means that the thousand Unit digit must not be $0$.)
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ :
(A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.
A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?
Each two-digit natural number is [i]assigned [/i] a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to $32$ is $6$ since $3 \times = 6$; the digit assigned to $93$ is $4$ since $9 \times 3 = 27$, $2 \times 7 = 14$, $1 \times 4 = 4$. Find all the two-digit numbers that are assigned $8$.
Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$.
[i](Proposed by Walther Janous)[/i]
The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen?
(F.L. Nazarov, Leningrad)