Found problems: 159
Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.
Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.
The natural number $N$ is a multiple of $2009$ and the sum of its (decimal) digits equals $2009$.
(a) Find one such number.
(b) Find the smallest such number.
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
Peter added a positive integer $M{}$ to a positive integer $N{}$ and noticed that the sum of the digits of the resulting integer is the same as the sum of the digits of $N{}$. Then he added $M{}$ to the result again, and so on. Will Peter eventually get a number with the same digit sum as the number $N{}$ again?
We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?
Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$?
[i]Proposed by Oleksii Masalitin[/i]
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]
Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$
for all $x,y\in \mathbb{N}$.
Note: $s(x)$ denotes the sum of digits of $x$.
[i]Proposed by Şevket Onur YILMAZ[/i]
Let us denote with $S(n)$ the sum of all the digits of $n$.
a) Is there such an $n$ that $n+S(n)=1980$?
b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.
Ann wrote down all the perfect squares from one to one million (all in a single line). However, at night, an evil elf erased one of the numbers. So the next day, Ann saw an empty space between the numbers $760384$ and $763876$. What is the sum of the digits of the erased number?
For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$.
Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.
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]
Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$
Find a positive integer $x$ such that the sum of the digits of $x$ is greater than $2011$ times the sum of the digits of the number $3x$ ($3$ times $x$).
Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$.
Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.
In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded)
\[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \]
as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.
It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.
Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its 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]
Let $n = 2^{p-1} (2^p - 1)$, and let $2^p- 1$ be a prime number. Prove that the sum of all (positive) divisors of $n$ (not including $n$ itself) is exactly $n$.
We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
Find the smallest positive integer $N$ satisfies :
1 . $209$│$N$
2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )