Found problems: 788
A set of five-digit numbers $\{N_1, ...,N_k\}$ is such that any five-digit number, all of whose digits are in non-decreasing order, coincides in at least one digit with at least one of the numbers $N_1$, $...$ , $N_k$. Find the smallest possible value of $k$.
Find the least positive integer $n$ with the property:
Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .
Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.
Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.
The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is
(A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above.
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$.
a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers.
b) Does there exist a positive integer such that each of its multiples is typical?
Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?
Show that there are infinitely many odd positive integers $n$ such that in binary $n$ has more $1$s than $n^2$.
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
Let $x, y$ be two positive integers, with $x> y$, such that $2n = x + y$, where n is a number two-digit integer. If $\sqrt{xy}$ is an integer with the digits of $n$ but in reverse order, determine the value of $x - y$
Find all $4$ digit number $\overline{abcd}$ such that $4\cdot \overline{abcd}+30=\overline{dcba}$
The decimal representation of an integer uses only two different digits. The number is at least $10$ digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?
Determine the sum of all positive integers $n$ that satisfy the following condition:
when $6n + 1$ is written in base $10$, all its digits are equal.
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 $0,1, \ldots, 9$.
Let $A, B, C$ and $D$ denote the digits in a four-digit number $n = ABCD$. Determine the least $n$ greater than $2017$ satisfying that there exists an integer $x$ such that $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$
For every non-negative integer $i$, define the number $M(i)$ as follows:
write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ )
(a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $.
Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ .
(b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ .
Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$.
(A Kanel)
We consider all $14$-digit positive integers, divisible by $18$, whose digits are exclusively $ 1$ and $2$, but there are no consecutive digits $2$. How many of these numbers are there?
Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.
Find all positive integers of form abcd such that
$$\overline{abcd} = a^{a+b+c+d} - a^{-a+b-c+d} + a$$
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.
Answer the following two questions and justify your answers:
(a) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+5^{2012}$?
(b) What is the last digit of the sum $1^{2012}+2^{2012}+3^{2012}+4^{2012}+...+2011^{2012}+2012^{2012}$?
Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.
We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order).
a) Determine the largest special number $m$ whose sum of digits is equal to $2023$.
b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.