How many three-digit numbers $\underline{abc}$ have the property that when it is added to $\underline{cba}$, the number obtained by reversing its digits, the result is a palindrome? (Note that $\underline{cba}$ is not necessarily a three-digit number since before reversing, $c$ may be equal to $0$.)

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?

Inés chose four different digits from the set $\{1,2,3,4,5,6,7,8,9\}$. He formed with them all possible four-digit numbers and added all those four-digit numbers. The result is $193314$. Find the four digits Inés chose.

Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit? Note: solved [url=https://artofproblemsolving.com/community/c6h312638p1685546]here[/url]

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

The infinite decimal notation of the real number $x$ contains all the digits. Let $u_n$ be the number of different $n$-digit segments encountered in $x$ notation. Prove that if for some $n$, $u_n \le (n+8)$, than $x$ is a rational number.

In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence? Folklore

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.$

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.

We take $100$ consecutive natural numbers $a_{1},$ $a_{2},$ $...,$ $a_{100}.$ Determine the last two digits of the number $a_{1}^{8}+a_{2}^{8}+...+a_{100}^{8}.$

A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$. [i]Proposed by Levi Iszler[/i]

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

How many numbers of $3$ digits have their central digit greater than any of the other two? How many of them have also three different digits?

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

Nils has a telephone number with eight different digits. He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?

A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that $$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: [i]the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of $13$, and the second digit of them is the average of the other two.[/i] What are the bus lines that go to Alberto's house?

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.

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?