Javier multiplies four digits, not necessarily different, and obtains a number ending in $7$. Determine how much the sum of the four digits that Javier multiplies can be worth. Give all the possibilities.

Starting from the number $ 1$ we write down a sequence of numbers where the next number in the sequence is obtained from the previous one either by doubling it, or by rearranging its digits (not allowing the first digit of the rearranged number to be $0$). For instance we might begin: $$1, 2, 4, 8, 16, 61, 122, 212, 424,...$$ Is it possible to construct such a sequence that ends with the number $1,000,000,000$? Is it possible to construct one that ends with the number $9,876,543,210$?

Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that: (a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds; (b) the above inequality is an equality for infinitely many positive integers, and (c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$ goes to zero as $ i$ goes to $ \infty.$ [i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$ (d) $ \infty;$ (e) an arbitrary real number $ \gamma \in (0,1)$; (f) an arbitrary real number $ \gamma \geq 0$; as $ i$ goes to $ \infty.$

Find two three-digit numbers $x$ and $y$ such that the sum of all other three digit numbers is equal to $600x$.

Call a year [i]ultra-even[/i] if all of its digits are even. Thus $2000,2002,2004,2006$, and $2008$ are all [i]ultra-even[/i] years. They are all $2$ years apart, which is the shortest possible gap. $2009$ is not an [i]ultra-even[/i] year because of the $9$, and $2010$ is not an ultra-even year because of the $1$. (a) In the years between the years $1$ and $10000$, what is the longest possible gap between two [i]ultra-even[/i] years? Give an example of two ultra-even years that far apart with no [i]ultra-even[/i] years between them. Justify your answer. (b) What is the second-shortest possible gap (that is, the shortest gap longer than $2$ years) between two [i]ultra-even[/i] years? Again, give an example, and justify your answer.

Does there exist an integer $A$ such that each of the ten digits $0, 1, . . . , 9$ appears exactly once as a digit in exactly one of the numbers $A, A^2, A^ 3$ ?

Oleksiy writes all the digits from $0$ to $9$ on the board, after which Vlada erases one of them. Then he writes $10$ nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these $10$ numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada? [i]Proposed by Oleksii Masalitin[/i]

Find all natural numbers $n$ with the following property: Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.

A positive integer $N$ is called [i]northern[/i] if for each digit $d > 0$, there exists a divisor of $N$ whose last digit is $d$. How many [i]northern [/i] numbers less than $2016$ are there with the fewest number of divisors as possible?

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

In a bank, only the manager knows the safe's combination, which is a five-digit number. To support this combination, each of the bank's ten employees is given a five-digit number. Each of these backup numbers has in one of the five positions the same digit as the combination and in the other four positions a different digit than the one in that position in the combination. Backup numbers are: $07344$, $14098$, $27356$, $36429$, $45374$, $52207$, $63822$, $70558$, $85237$, $97665$. What is the combination to the safe?

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

An eight-digit number is said to be 'robust' if it meets both of the following conditions: (i) None of its digits is $0$. (ii) The difference between two consecutive digits is $4$ or $5$. Answer the following questions: (a) How many are robust numbers? (b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all the super robust numbers.

The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition: [i]for any two different digits from $1,2,3,4,,6,7,8$ there exists a number in $X$ which contains both of them. [/i]\\ Determine the smallest possible value of $N$.

On blackboard is written $3$ digit number so all three digits are distinct than zero. Out of it, we made three $2$ digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is $293$. Which number is written on blackboard?

Delete $100$ digits from the number $1234567891011... 9899100$ so that the remaining number were as big as possible.

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.

Prove that the second last digit of each power of three is even . (V . I . Plachkos)

A number is said to be [i]balanced [/i] if one of its digits is average of the others. How many [i]balanced [/i]$3$-digit numbers are there?

Find the sum of all $4$-digit numbers using the digits $2,3,4,5,6$ without a repetition of any of those 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?

Let $a$ and $b$ be positive integers such that (i) both $a$ and $b$ have at least two digits; (ii) $a + b$ is divisible by $10$; (iii) $a$ can be changed into $b$ by changing its last digit. Prove that the hundreds digit of the product $ab$ is even.

Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.

Danny likes seven-digit numbers with the following property: the 1's digit is divisible by the 10's digit, the 10's digit is divisible by the 100's digit, and so on. For example, Danny likes the number $1133366$ but doesn't like $9999993$. Is the amount of numbers Danny likes divisible by $7$?