Found problems: 788
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.
Consider a sequence of non-negative integers g$_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$, for example, is written as $027$). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$, $g_3= 015$, $g_4 = 000$).Prove that independent of the choice of $g_1$:
(a) $g_{n+1}\le g_n$
(b) $g_{10}= 000$.
For the positive integer $k$ we denote by the $a_n$ , the $k$ from the left digit in the decimal notation of the number $2^n$ ($a_n = 0$ if in the notation of the number $2^n$ less than the digits). Consider the infinite decimal fraction $a = \overline{0, a_1a_2a_3...}$. Prove that the number $a$ is irrational.
The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?
A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.
Find all prime numbers with $n \ge 3$ digits, having the property:
for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.
Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$.
[i]Proposed by Oleksiy Masalitin[/i]
Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?
Determine all positive integers $n$ such that the decimal representation of the number $6^n + 1$ has all its digits the same.
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both $a/b$ and $b/a$), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.
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 $S(m)$?
A $2015$- digit natural number $A$ has the property that any $5$ of it's consecutive digits form a number divisible by $32$. Prove that $A$ is divisible by $2^{2015}$
Find all the natural $n$ and $k$ such that $n^n$ has $k$ digits and $k^k$ has $n$ digits.
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.
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.
Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$
Let $c_n$ be the first digit of $2^n$ (in decimal representation). Prove that the number of different $13$-tuples $< c_k$,$...$, $c_{k+12}>$ is equal to $57$.
(AY Belov,)
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.
In the decimal expansion of a fraction $\frac{m}{n}$ with positive integers $m$ and $n$ you can find a string of numbers $7143$ after the comma.
Show $n>1250$.
[i]Example:[/i] I mean something like $0.7143$.
Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?
How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ?
(A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.
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 $n$.
A natural number is called a palindrome if it is read in the same way. from left to right and from right to left (in particular, the last digit of the palindrome coincides with the first and therefore not equal to zero). Squares of two different natural numbers have $1001$ digits. Prove that strictly between these squares, there is one palindrome.
Consider the positive integers written in the decimal system with $n$ digits, the start of which is not zero and where there are no two sevens next to each other. The number of these numbers is called $u_n$. Derive a relation that expresses $u_{n+2}$ in terms of $u_{n+1}$ and $u_n$.