Found problems: 788
Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?
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.
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
The digits of the $17$-digit number are rearranged in the reverse order. Prove that at list one digit of the sum of the new and the initial number is even.
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 $n \geq n_0, d_{n+T} = d_n.$
The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once.
(a) Find the smallest positive difference $d$ of two numbers from $M$.
(b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$?
(Gerhard Kirchner)
Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.
We are looking for numbers ending with the digit $5$ such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property.
(a) Find four such numbers.
(b) Prove that there are infinitely many.
(A. Andjans, Riga)
Find all solutions of the system of equations
$\begin{cases} s(x) + s(y) = x \\ x + y + s(z) = z \\ s(x) + s(y) + s(z) = y - 4 \end{cases}$
where $x, y$, and $z$ are positive integers, and $s(x), s(y)$, and $s(z)$ are the numbers of digits in the decimal representations of $x, y$, and $z$, respectively.
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.
There is a positive integer $A$. Two operations are allowed: increasing this number by $9$ and deleting a digit equal to $1$ from any position. Is it always possible to obtain $A+1$ by applying these operations several times?
We call S $(n)$ the sum of the digits of the integer $n$. For example, $S (327)=3+2+7=12$.
Find the value of $$A=S(1)-S(2)+S(3)-S(4)+...+S(2011)-S(2012).$$
($A$ has $2012$ terms).
Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.
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)
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$
and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for
all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.
For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n)=3 \cdot S(2n)$.
Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.
We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.
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 \[f(x)=138\,012.\]
Prove that every positive integer has a multiple whose decimal representation involves all ten digits.
Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?
Find the smallest $3$-digit number that is the product of two $2$-digit numbers , so that the seven digits of these three numbers are all different.
$72$ 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.
Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.