Found problems: 788
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:
(i) $n$ has exactly $k$ digits (in decimal representation),
(ii) all the digits of $n$ are odd,
(iii) $n$ is divisible by $5$,
(iv) the number $m = n/5$ has $k$ odd digits
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that
(a) The sum of the digits of $2M$ equals the sum of the digits of $2K$.
(b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even).
(c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$.
(AD Lisitskiy)
Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.
Find the number of five-digit numbers whose square ends in the same five digits in the same order.
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
Find any two consecutive natural numbers, each of which is divisible by the square of the sum of its digits.
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$.
For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.
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.
Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$
What is the last two digits of the number $(11 + 12 + 13 + ... + 2006)^2$?
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:
(i) move the last digit of $a$ to the first position to obtain the numb er $b$;
(ii) square $b$ to obtain the number $c$;
(iii) move the first digit of $c$ to the end to obtain the number $d$.
(All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)
Find all numbers $a$ for which $d\left( a\right) =a^2$.
[i]Proposed by Zoran Sunic, USA[/i]
Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that
\begin{align*}
n+s(2n)=m+s(2m) \\
kn+s(n^2)=km+s(m^2).
\end{align*}
($s(n)$ denotes the sum of digits of $n$.)
[i]Proposed by Mohammadamin Sharifi[/i]
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 $10$.
Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?
Prove that for every natural $n$ there exists a number, containing only digits "$1$" and "$2$" in its decimal notation, that is divisible by $2^n$ ( $n$-th power of two ).
Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.
Juan wrote a natural number and Maria added a digit $ 1$ to the left and a digit $ 1$ to the right. Maria's number exceeds to the number of Juan in $14789$. Find the number of Juan.
How many positive integers less than $10,000$ have an even number of even digits and an odd number of odd digits ?
(Assume no number starts with zero.)
Find $5$ last digits of the number $M = 5^{2010}$ .
(A): $65625$, (B): $45625$, (C): $25625$, (D): $15625$, (E) None of the above.
Prove that there exists a positive integer $n$ such that the last digits of $n^3$ are $...201320132013$.
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 $$2/5 \le m/n \le 3/5$$
How many numbers $\overline{abcd}$ with different digits satisfy the following property:
if we replace the largest digit with the digit $1$ results in a multiple of $30$?
The numbers $25$ and $76$ have the property that when squared in base $10$, their squares also end in the same two digits. A positive integer is called [i]amazing [/i] if it has at most $3$ digits when expressed in base $21$ and also has the property that its square expressed in base $21$ ends in the same $3$ digits. (For this problem, the last three digits of a one-digit number b are 00b, and the last three digits of a two-digit number $\underline{ab}$ are $0\underline{ab}$.) Compute the sum of all amazing numbers. Express your answer in base $21$.