Found problems: 573
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.
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 $u_n$.
Determine the number of positive integers $n$ satisfying:
[list]
[*] $n<10^6$
[*] $n$ is divisible by 7
[*] $n$ does not contain any of the digits 2,3,4,5,6,7,8.
[/list]
We call a positive integer a [i]shuffle[/i] number if the following hold:
(1) All digits are nonzero.
(2) The number is divisible by $11$.
(3) The number is divisible by $12$. If you put the digits in any other order, you again have a number that is divisible by $12$.
How many $10$-digit [i]shuffle[/i] numbers are there?
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.)
A six-digit number is [i]balanced [/i] when all digits are different from zero and the sum of the first three digits is equal to the sum of the last three digits. Prove that the sum of all six-digit balanced numbers is divisible by $13$.
How many ways are there to insert plus signs $+$ between the digits of number $111111 ...111$ which includes thirty of digits $1$ so that the result will be a multiple of $30$?
What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$?
(M . I. Gusarov)
Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.
Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$. How many passcodes satisfy these conditions?
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.)
How many five digit numbers are divisible by $3$ and contain the digit $6$?
The number $x$ from $[0,1]$ is written as an infinite decimal fraction. Having rearranged its first five digits after the point we can obtain another fraction that corresponds to the number $x_1$. Having rearranged five digits of $x_k$ from $(k+1)$-th till $(k+5)$-th after the point we obtain the number $x_{k+1}$.
a) Prove that the sequence $x_i$ has limit.
b) Can this limit be irrational if we have started with the rational number?
c) Invent such a number, that always produces irrational numbers, no matter what digits were transposed.
The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether?
(G. Galperin)
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
For $n\in\mathbb{N}$, prove that $2^n$ can begin with any sequence of digits.
Hint: $\log 2$ is irrational number.
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$
Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits?
(S Tokarev)
Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $2000$ digits?
Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.
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 digits of the $17$-digit number are rearranged in the reverse order. Prove that at list one digit of the sum of the new and the initial number is even.
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 $n \geq n_0, d_{n+T} = d_n.$