Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

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.

What is the last two digits of the number $(11 + 12 + 13 + ... + 2006)^2$?

Does there exist a $6$-digit number $A$ such that none of its $500 000$ multiples $A$, $2A$, $3A$, ..., $500 000A$ ends in $6$ identical digits? (S Tokarev)

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.

How many $0$'s are there in the sequence $x_1, x_2,..., x_{2014}$ where $x_n =\big[ \frac{n + 1}{\sqrt{2015}}\big] -\big[ \frac{n }{\sqrt{2015}}\big]$ , $n = 1, 2,...,2014$ ? (A): $1128$, (B): $1129$, (C): $1130$, (D): $1131$, (E) None of the above.

(a) Decide whether there exist two decimal digits $a$ and $b$, such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$. (b) Decide whether there exist two decimal digits $c$ and $d$, such that every integer with decimal representation $cd222... 231$ is divisible by $79$.

Find all prime numbers of the form $10101...01$.

Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.

Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$

Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

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.

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

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

Consider positive integers $m$ and $n$ for which $m> n$ and the number $22 220 038^m-22 220 038^n$ has are eight zeros at the end. Show that $n> 7$.

Victor writes down all $7-$digit numbers using the digits $1, 2, 3, 4, 5, 6,$ and $7$ exactly once. Prove that there are no two numbers among them where one is a multiple of the other.

All the $5$-digit numbers from $11111$ to $99999$ are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting $444445$-digit number is not a power of two.

How many positive integers $N$ in the segment $\left[10, 10^{20} \right]$ are such that if all their digits are increased by $1$ and then multiplied, the result is $N+1$? [i](F. Bakharev)[/i]

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?

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 $9,876,543,210$?

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?