Found problems: 128
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 digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
Find digits $x, y, z$ such that the equality
\[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\]
holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.
Find the smallest natural number $n$ which has the following properties:
a) Its decimal representation has a 6 as the last digit.
b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number $n$.
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.
Prove that the first 3 digits after the decimal point in the decimal expression of the number $\frac{0.123456789101112 . . . 495051}{0.515049 . . . 121110987654321}$ are $239$.
Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.
(E Turkevich)
Show that there is a multiple of $2^{1998}$ whose decimal representation consists only of the digits $1$ and $2$.
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.
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]
The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.
Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$.
Can Alice avoid that?
[i](Proposed by Richard Henner)[/i]
Let $f(k)$ be the number of integers $n$ satisfying the following conditions:
(i) $0\leq n < 10^k$ so $n$ has exactly $k$ digits (in decimal notation), with leading zeroes allowed;
(ii) the digits of $n$ can be permuted in such a way that they yield an integer divisible by $11$.
Prove that $f(2m) = 10f(2m-1)$ for every positive integer $m$.
[i]Proposed by Dirk Laurie, South Africa[/i]
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]