Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.

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

$7.$ Let $E(n)$ denote the sum of even digits of $n.$ For example, $E(1243)=2+4=6.$ What is the value of $E(1)+E(2)+E(3)+...+E(100) ?$

Let $a$, $b$ $c$ be the natural numbers, such that every digit occurs exactly the same number of times in each of the numbers $a$, $b$, $c$. Is it possible that $a + b + c = 10^{1001}$? Justify your answer.

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.

Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$. $\bullet$ $\overline{abc}$ is a multiple of $36$. $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers.

Find (with proof) all natural numbers $n$ such that, for some natural numbers $a$ and $b$, $a\ne b$, the digits in the decimal representations of the two numbers $n^a+1$ and $n^b+1$ are in reverse order.

Determine the number of ten-digit positive integers with the following properties: $\bullet$ Each of the digits $0, 1, 2, . . . , 8$ and $9$ is contained exactly once. $\bullet$ Each digit, except $9$, has a neighbouring digit that is larger than it. (Note. For example, in the number $1230$, the digits $1$ and $3$ are the neighbouring digits of $2$ while $2$ and $0$ are the neighbouring digits of $3$. The digits $1$ and $0$ have only one neighbouring digit.) [i](Karl Czakler)[/i]

Given five $n$-digit binary numbers. For each two numbers their digits coincide exactly on $m$ places. There is no place with the common digit for all the five numbers. Prove that $$2/5 \le m/n \le 3/5$$

In a sequence of positive integers an inversion is a pair of positions such that the element in the position to the left is greater than the element in the position to the right. For instance the sequence $2,5,3,1,3$ has five inversions - between the first and fourth positions, the second and all later positions, and between the third and fourth positions. What is the largest possible number of inversions in a sequence of positive integers whose sum is $2014$?

Find all positive integers $\overline{xyz}$ ($x$, $y$ and $z$ are digits) such that $\overline{xyz} = x+y+z+xy+yz+zx+xyz$

Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

The numbers $2^{1989}$ and $5^{1989}$ are written out one after the other (in decimal notation). How many digits are written altogether? (G. Galperin)

A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$. If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.

Given a positive integer $N$, define $u(N)$ as the number obtained by making the ones digit the left-most digit of $N$, that is, taking the last, right-most digit (the ones digit) and moving it leftwards through the digits of $N$ until it becomes the first (left-most) digit; for example, $u(2023) = 3202$. [b](1)[/b] Find a $6$-digit positive integer $N$ such that \[\frac{u(N)}{N} = \frac{23}{35}.\] [b](2)[/b] Prove that there is no positive integer $N$ with less than $6$ digits such that \[\frac{u(N)}{N} = \frac{23}{35}.\]

What are last three digits of $2^{2^{2004}}$ ?

For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime? [i]Proposed by Oleksii Masalitin[/i]

A positive integer is called [i]vaivém[/i] when, considering its representation in base ten, the first digit from left to right is greater than the second, the second is less than the third, the third is bigger than the fourth and so on alternating bigger and smaller until the last digit. For example, $2021$ is [i]vaivém[/i], as $2 > 0$ and $0 < 2$ and $2 > 1$. The number $2023$ is not [i]vaivém[/i], as $2 > 0$ and $0 < 2$, but $2$ is not greater than $3$. a) How many [i]vaivém[/i] positive integers are there from $2000$ to $2100$? b) What is the largest [i]vaivém[/i] number without repeating digits? c) How many distinct $7$-digit numbers formed by all the digits $1, 2, 3, 4, 5, 6$ and $7$ are [i]vaivém[/i]?

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)

How many words with $n$ digits can be formed from the alphabet $\{0, 1, 2, 3, 4\}$, if neighboring digits must differ by exactly one? [i]Proposed by Germany, FR.[/i]

A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

Call a positive integer [i]balanced [/i] if the number of its distinct prime factors is equal to the number of its digits in the decimal representation; for example, the number $385 = 5 \cdot 7 \cdot 11$ is balanced, while $275 = 5^2 \cdot 11$ is not. Prove that there exist only a finite number of balanced numbers.