Found problems: 788
Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.
Prove that every integer $k > 1$ has a multiple less than $k^4$ whose decimal expension has at most four distinct digits.
For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime?
[i]Proposed by Oleksii Masalitin[/i]
We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits?
(Paolo Leonetti)
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.
In the sequence of powers of $2$ (written in the decimal system, beginning with $2^1 = 2$) there are three terms of one digit, another three of two digits, another three of $3$, four out of $4$, three out of $5$, etc. Clearly reason the answers to the following questions:
a) Can there be only two terms with a certain number of digits?
b) Can there be five consecutive terms with the same number of digits?
c) Can there be four terms of n digits, followed by four with $n + 1$ digits?
d) What is the maximum number of consecutive powers of $2$ that can be found without there being four among them with the same number of digits?
A toymaker has $k$ dice at his disposal, each with $6$ blank sides. On each side of each of these dice, the toymaker must draw one of the digits $0, 1, 2, \ldots , 9$.
Determine (in terms of $k$) the largest integer $n$ such that the toymaker can draw digits on the $k$ dice such that, for any positive integer $r \leq n$, it is possible to choose some of the $k$ dice and form with them the decimal representation of $r$.
[b]Note:[/b] The digits 6 and 9 are distinguishable: they appear as [u]6[/u] and [u]9[/u].
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.
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?
All natural numbers containing not more than $n$ digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if $0<k<n$ than the sum of the $k$-th powers of the numbers in the first group equals to the sum of the $k$-th powers of the numbers in the second group.
We say that a positive integer $M$ with $2n$ digits is [i]hypersquared[/i] if the following three conditions are met:
[list]
[*]$M$ is a perfect square.
[*]The number formed by the first $n$ digits of $M$ is a perfect square.
[*]The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero).
[/list]
Find a hypersquared number with $2000$ digits.
A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.
Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
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]
The positive integer $a$ is relatively prime with $10$. Prove that for any positive integer $n$, there exists a power of $a$ whose last $n$ digits are $\underbrace{0...0}_\text{n-1}1$.
Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.
Find the sum of all the $5$-digit integers which are not multiples of $11$ and whose digits are $1, 3, 4, 7, 9$.
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
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 $a,b<10^{10}$.
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$.
a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers.
b) Does there exist a positive integer such that each of its multiples is typical?
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a]
[*] the square of this number;
[*]the fourth power of this number,
[/list] given that $k\geqslant 4.$
[i]From the folklore[/i]