Found problems: 573
How many five-digit multiples of 3 end with the digit 6 ?
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
We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.
There are many sets of two different positive integers $a$ and $b$, both less than $50$, such that $a^2$ and $b^2$ end in the same last two digits. For example, $35^2 = 1225$ and $45^2 = 2025$ both end in $25$. What are all possible values for the average of $a$ and $b$?
For the purposes of this problem, single-digit squares are considered to have a leading zero, so for example we consider $2^2$ to end with the digits 04, not $4$.
Which positive integers satisfy that the sum of the number’s last three digits added to the number itself yields $2029$?
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$
Let $A$ be an even number but not divisible by $10$. The last two digits of $A^{20}$ are:
(A): $46$, (B): $56$, (C): $66$, (D): $76$, (E): None of the above.
Is there a number whose digits are only $1$'s and which is divided by $1999$?
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.
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?
Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times.
(A Shapovalov)
$a$ is a $2000$ digit natural number of the form
$$a=2(A)99…99(B)(C)$$
expressed in base $10$. $a$ is not a multiple of $10$, and $2(A)+(B)(C)=99$. $a=2899..9971$ is a possible example of $a$. $b$ is a number you earn when you write the digits of $a$ in a reverse order(Writing the digits of some number in a reverse order means like reordering $1234$ into $4321$). Find every positive integer $a$ that makes $ab$ a square number.
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values of $N $.
Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?
Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$.
a) Would the monster have woken up between the years $2005$ and $2015$?
b) Will we be safe in the next $10$ years?
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?
Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$.
(Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)
Determine, with proof, the smallest positive multiple of $99$ all of whose digits are either $1$ or $2$.
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 $n$ 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.
Is there a nine-digit number without zero digits, the remainder of dividing which on each of its digits is different?