Found problems: 788
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints).
[b]a)[/b] Which digit did Sophie write the most?
[b]b)[/b] Which digit did Sophie write the least?
Prove that none of the digits $2$, $4$, $7$, $9$ can be the last digit of a number
$$
1 + 2 + 3 + \ldots + n,$$
where $n$ is a natural number.
Find the smallest positive integer $N$ satisfies :
1 . $209$│$N$
2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )
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?
A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.
$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .
Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$.
Find the first $200$ decimal digits for the number $a^{2000}$.
A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?
Is it possible to select two natural numbers $m$ and $n$ so that the number $n$ results from a permutation of the digits of $m$, and $m+n =999 . . . 9$ ?
All $10$-digit numbers consisting of digits $1, 2$ and $3$ are written one under the other. Each number has one more digit added to the right. $1$, $2$ or $3$, and it turned out that to the number $111. . . 11$ added $1$ to the number $ 222. . . 22$ was assigned $2$, and the number $333. . . 33$ was assigned $3$. It is known that any two numbers that differ in all ten digits have different digits assigned to them. Prove that the assigned column of numbers matches with one of the ten columns written earlier.
From digits $0$, $1$, $3$, $4$, $7$ and $9$ were written $5$ digit numbers which all digits are different. How many numbers from them are divisible with $5$
Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.
$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$
Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?
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 $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
Find all four-digit natural numbers formed by two even digits and two odd digits that verify that when multiplied by $2$ four-digit numbers are obtained with all their even digits and when divided by $2$ four-digit natural numbers are obtained with all their odd digits.
For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$. The infinite sequence of these digits starts with $2,6,4,2,2$. Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.
Let $f: Z^+ \to Z^+ \cup \{0\}$ a function that meets the following conditions:
a) $f (a b) = f (a) + f (b)$,
b) $f (a) = 0$ provided that the digits of the unit of $a$ are $7$,
c) $f (10) = 0$.
Find $f (2016).$
How many $8$-digit numbers are $*2*0*1*7$, where four unknown numbers are replaced by stars, which are divisible by $7$?
The base-$7$ representation of number $n$ is $\overline{abc}_{(7)}$, and the base-$9$ representation of number $n$ is $\overline{cba}_{(9)}$. What is the decimal (base-$10$) representation of $n$?
What are last three digits of $2^{2^{2004}}$ ?
The decimal notation of three natural numbers consists of equal digits: $n$ digits $x$ for $a$, $n$ digits $y$ for $b$ and $2n$ digits $z$ for $c$. For every $n > 1$ find all the possible triples of digits $x,y,z$ such, that $a^2 + b = c$
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
How many numbers greater than $1.000$ but less than $10.000$ have as a product of their digits $256$?