Found problems: 788
For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions
$$a \mid b + s(a)$$
$$b \mid a + s(b)$$
[i]Proposed by Victor Domínguez[/i]
The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.
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]
Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.
Consider the natural numbers written in the decimal system.
(a) Find the smallest number which decreases five times when its first digit is erased. Which form do all numbers with this property have?
(b) Prove that there is no number that decreases $12$ times when its first digit is erased.
(c) Find the necessary and sufficient condition on $k$ for the existence of a natural number which is divided by $k$ when its first digit is erased.
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 $9,876,543,210$?
Does there exist a four-digit positive integer with different non-zero digits, which has the following property: if we add the same number written in the reverse order, then we get a number divisible by $101$?
Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$.
Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.
An eight-digit number is said to be 'robust' if it meets both of the following conditions:
(i) None of its digits is $0$.
(ii) The difference between two consecutive digits is $4$ or $5$.
Answer the following questions:
(a) How many are robust numbers?
(b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all
the super robust numbers.
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find
\[
s_1 + s_2 + \cdots + s_n
\]
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]
For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let $f(m)=m+S(m)$.
Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$
Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set
$\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.
Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.
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 the largest integer $d$ divides all three numbers $abc, bca$ and $cab$ with $a, b$ and $c$ being some nonzero and mutually different digits.
Czech Republic
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]
Let $n$ be an integer. We consider $s (n)$, the sum of the $2001$ powers of $n$ with the exponents $0$ to $2000$. So $s (n) = \sum_{k=0}^{2000}n ^k$ . What is the unit digit of $s (n)$ in the decimal system?
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
Show that the $n$ digits after the decimal point in $(5 +\sqrt{26})^n$ are all equal.
Let $a^b$ denote the number $ab$. The order of operations in the expression 7^7^7^7^7^7^7 must be determined by parentheses ($5$ pairs of parentheses are needed). Is it possible to put parentheses in two distinct ways so that the value of the expression be the same?