Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation. That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

Find, with proof, the least positive integer $n$ having the following property: in the binary representation of $\frac 1n$, all the binary representations of $1, 2, \ldots, 1990$ (each consist of consecutive digits) are appeared after the decimal point.

Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

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.

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.

Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$. (Proposed by Gerhard Woeginger, Austria)

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

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.

Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]

Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write \[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \] where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\] a) Prove that $S$ is infinite. b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$

When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

Which fractions $ \dfrac{p}{q},$ where $p,q$ are positive integers $< 100$, is closest to $\sqrt{2} ?$ Find all digits after the point in decimal representation of that fraction which coincide with digits in decimal representation of $\sqrt{2}$ (without using any table).

[b]1.[/b] For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$. a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$. b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$.

Let $n\geq 2$ be a positive integer. Prove that there exists a positive integer $m$, such that $n\mid m, \ m<n^4$ and at most four distinct digits are used in the decimal representation of $m$.

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.

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

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 $d_n$ 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 $n \geq n_0, d_{n+T} = d_n.$

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

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 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 $n$.

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 $n \geq n_0, d_{n+T} = d_n.$