Found problems: 788
Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by
permuting these digits leaves the same remainder after the division by $4$.
A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.
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]
Given the number $188188...188$ (number $188$ is written $101$ times). Some digits of this number are crossed out. What is the largest multiple of $7$, that could happen?
To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.
Find the smallest four-digit number that when divided by $3$ gives a four-digit number with the same digits.
(Note: Four digits means that the thousand Unit digit must not be $0$.)
How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ :
(A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.
Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$
[i]The Problem Selection Committee[/i]
On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.
A positive integer is called special if its digits can be arranged to form an integer divisible by $4$. How many of the integers from $1$ to $2018$ are special?
Each two-digit natural number is [i]assigned [/i] a digit as follows: Its digits are multiplied. If the result is a digit, this is the assigned digit. If the result is a two-digit number, these two figures are multiplied, and if the result is a digit, this is the assigned digit. Otherwise, the operation is repeated. For example, the digit assigned to $32$ is $6$ since $3 \times = 6$; the digit assigned to $93$ is $4$ since $9 \times 3 = 27$, $2 \times 7 = 14$, $1 \times 4 = 4$. Find all the two-digit numbers that are assigned $8$.
Determine all digits $z$ such that for each integer $k \ge 1$ there exists an integer $n\ge 1$ with the property that the decimal representation of $n^9$ ends with at least $k$ digits $z$.
[i](Proposed by Walther Janous)[/i]
The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen?
(F.L. Nazarov, Leningrad)
How many five-digit multiples of 3 end with the digit 6 ?
In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?
Folklore
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 $A'B<AB'$.
There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^2$ and $b^2$ end in the same last two digits. For example, $35^2 = 1225$ and $45^2 = 2025$ both end in $25$. What are all possible values for the average of $a$ and $b$?
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^2$ to end with the digits 04, not $4$.
Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$
Let $A$ be an even number but not divisible by $10$. The last two digits of $A^{20}$ are:
(A): $46$, (B): $56$, (C): $66$, (D): $76$, (E): None of the above.
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]
How many zeros does $100!$ have at its end in the usual decimal representation?
Is there a number whose digits are only $1$'s and which is divided by $1999$?
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
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]