Found problems: 573
Let $A$ and $B$ be the number of odd positive integers $n<1000$ for which the number formed by the last three digits of $n^{2015}$ is greater and smaller than $n$, respectively. Prove that $A=B$.
Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?
We are given two two-digit numbers , $x$ and $y$. It is known that $x$ is twice as big as $y$. One of the digits of $y$ is the sum, while the other digit of $y$ is the difference, of the digits of $x$ . Find the values of $x$ and $y$, proving that there are no others.
We say that a four-digit number $\overline{abcd}$ ($a \ne 0$) is [i]pora [/i] if the following terms are true :
$\bullet$ $a\ge b$
$\bullet$ $ab - cd = cd -ba$.
For example, $2011$ is pora because $20-11 = 11-02$
Find all the numbers around.
Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.
A schoolgirl forgot to write a multiplication sign between two $3$-digit numbers and wrote them as one number. This $6$-digit result proved to be $3$ times greater than the product (obtained by multiplication). Find these numbers.
(A Kovaldzhi,
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
\[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]
Let $\overline{ABCD}$ be a $4$-digit number. What is the smallest possible positive value of $\overline{ABCD}- \overline{DCBA}$?
Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times.
Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.
How many positive five-digit integers are there that have the product of their five digits equal to $900$?
(Karl Czakler)
Let $T$ be the smallest positive integers which, when divided by $11,13,15$ leaves remainders in the sets {$7,8,9$}, {$1,2,3$}, {$4,5,6$} respectively. What is the sum of the squares of the digits of $T$ ?
The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.
Determine the number of three-digit numbers with the following property:
The number formed by the first two digits is prime and the number formed by the last two digits is prime.
Find all four-digit numbers $\overline{abcd}$ (in decimal system) such that $\overline{abcd}= (\overline{ac}+1).(\overline{bd} +1)$
The decimal representation of $$\dfrac{1}{20^{20}}$$ consists of a string of zeros after the decimal point, followed by a 9 and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
$\textbf{(A) }23\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27$
For each positive integer $n$, let $c(n)$ be the number of digits of $n$. Let $A$ be a set of positive integers with the following property: If $a$ and $b$ are two distinct elements in $A$, then $c(a +b)+2 > c(a)+c(b)$. Find the largest number of elements that $A$ can have.
PS. In the original wording: c(n) = ''cantidad de dıgitos''
Sixteen natural numbers written in the decimal system may form a geometric sequence, of which the first five members have nine digits, five further members have ten digits, four members have eleven digits and two terms have twelve digits. Prove that there is exactly one sequence with these properties.
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$,
2. $N$ only has three different digits, and none of them are zero,
3. Exactly 99.9% of the digits of $N$ are odd.
Does there exist an integer containing only digits $2$ and $0$ which is a $k$-th power of a positive integer ($k \ge2$)?
A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$.
If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.
The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$.
(a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system.
(b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.
Elisa adds the digits of her year of birth and observes that the result coincides with the last two digits of the year her grandfather was born. Furthermore, the last two digits of the year she was born are precisely the current age of her grandfather. Find the year Elisa was born and the year her grandfather was born.
Let $X= a1b9$ and $Y ab = 51ab$ be two positive integers where $a$ and $b$ are digits. $X$ is known to be multiple of a positive two-digit number $n$ and $Y$ is the next multiple of that number $n$. Find the number $n$ and the digits $a$ and $b$. Justify why there are no other possibilities.
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)