Found problems: 788
The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?
Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.
Let $n$ be a natural number. Prove that there is a multiple of $n$ that can be written only with the digits $0$ and $1$.
Let us denote with $S(n)$ the sum of all the digits of $n$.
a) Is there such an $n$ that $n+S(n)=1980$?
b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.
We call a positive integer [i]alternating[/i] if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers $n$ such that $n$ has a multiple which is alternating.
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 $0,1, \ldots, 9$.
Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?
Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?
Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$.
[b](1)[/b] Find a $6$-digit positive integer $N$ such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
[b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that
\[\frac{u(N)}{N} = \frac{23}{35}.\]
For which natural $n$ there exists a natural number multiple of $n$, whose decimal notation consists only of the digits $8$ and $9$ (possibly only from numbers $8$ or only from numbers $9$)?
Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.
Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. The last digit of $S$ is
(A): $1$ (B): $3$ (C): $5$ (D): $7$ (E): None of the above.
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$.
Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
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.
Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?
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.
Pedro writes all the numbers with four different digits that can be made with digits $a, b, c, d$, that meet the following conditions: $$ a\ne 0 \, , \, b=a+2 \, , \, c=b+2 \, , \, d=c+2$$
Find the sum of all the numbers Pedro wrote.
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 $m$.
Find the sum of all $4$-digit numbers using the digits $2,3,4,5,6$ without a repetition of any of those digits.
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]
Find the smallest multiple of $81$ that only contains the digit $1$. How many $ 1$’s does it contain?
Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?
Consider the sequence of numbers defined by $a_1 = 7$, $a_2 = 7^7$ , $ ...$ , $a_n = 7^{a_{n-1}}$ for $n \ge 2$. Determine the last digit of the decimal representation of $a_{2021}$.
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$