Found problems: 788
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds:
$$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$
For which largest $k$ do there exist $k$-similar numbers?
[i]Proposed by Oleksiy Masalitin[/i]
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.
Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.
If $a$ and $b$ are digits, how many are there $4$ digit numbers $\overline{3ab4}$ divisible with $9$ . Which numbers are they ($4$ digit numbers)?
Let $N$ be a positive integer. Prove that at least one of the numbers $N$ of $3N$ contains at least one of the digits $1,2,9$.
[i]Proposed by Evan Chang (squareman), USA[/i]
Prove that the square of any integer cannot end with two fives.
Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote?
[i]Proposed by Oleksiy Masalitin[/i]
A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$,
$$a_{k+1}=3a_k^4+4a_k^3.$$
Prove that $a_{10}$ contains more than $1000$ nines in decimal notation.
(Yao)
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.
The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them.
For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums.
(E. Barabanov)
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which
\[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\]
where $s(n)$ denotes the sum of digits of $n$ in decimal representation.
Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.