Found problems: 573
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)
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?
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 \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
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.
In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?
Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number.
We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$.
For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$.
[list=a]
[*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number.
[*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$.
[*] Find the residue of $x(5776)-x(5708)\mod 2016$.
[/list]
How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)
A game for two.
One gives a digit and the second substitutes it instead of a star in the following difference:
$$**** - **** = $$
Then the first gives the next digit, and so on $8$ times.
The first wants to obtain the greatest possible difference, the second -- the least. Prove that:
1. The first can operate in such a way that the difference would be not less than $4000$, not depending on the second's behaviour.
2. The second can operate in such a way that the difference would be not greater than $4000$, not depending on the first's behaviour.
We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.
Positive integer $m$ shall be called [i]anagram [/i] of positive $n$ if every digit $a$ appears as many times in the decimal representation of $m$ as it appears in the decimal representation of $n$ also. Is it possible to find $4$ different positive integers such that each of the four to be [i]anagram [/i] of the sum of the other $3$?
(Bulgaria)
Are there $n$-digit numbers M and N such that all digits $M$ are even, all $N$ digits are odd, every digit from $0$ to $9$ occurs in decimal notation M or N at least once, and $M$ is divisible by $N$?
Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?
Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]
How many 4-digit numbers $ABCD$ are there with the property that $|A-B|= |B-C|= |C-D|$?
Note that the first digit $A$ of a four-digit number cannot be zero.