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 $$x =\sqrt{A +\sqrt{B +\sqrt{C +\sqrt{D + x}}}}.$$

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.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

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

We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero. a) Find a quad-divi number such that the sum of its digits is $24$. b) Find a quad-divi number such that the sum of its digits is $1001$.

For each positive integer $n$, a new number $t_n$ is formed from the numbers $2^n$ and $5^n$ which consists of the digits from $2^n$ followed by the digits from $5^n$. For example, $t_4$ is $16625$. How many digits does the number $t_{2008}$ have?

Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.

We will call a positive integer [i]good [/i] if all its digits are nonzero. A good integer will be called [i]special [/i] if it has at least $k$ digits and their values strictly increase from left to right. Let a good integer be given. At each move, one may either add some special integer to its digital expression from the left or from the right, or insert a special integer between any two its digits, or remove a special number from its digital expression.What is the largest $k$ such that any good integer can be turned into any other good integer by such moves?

* Prove that a number of the form $2^n$ for a positive integer $n$ may begin with any given combination of digits.

Find digits $a,b,c,x$ ($a>0$) such that $\overline{abc}+\overline{acb}=\overline{199x}$

We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$) is at position $1$. An integer is called [i]even-steven[/i] if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist). An integer is called [i]oddball[/i] if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist). For example, $3122$ is [i]oddball[/i] but not [i]even-steven[/i], $7$ is both [i]even-steven[/i] and [i]oddball[/i], and $123$ is neither [i]even-steven[/i] nor [i]oddball[/i]. (a) Prove: every oddball integer greater than $9$ can be obtained by adding two [i]oddball [/i] integers. (b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two [i]even-steven[/i] integers.

Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$