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]

Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

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]

Find all positive integers that are equal to $700$ times the sum of its digits.

Find any two consecutive natural numbers, each of which is divisible by the square of the sum of its digits.

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)

For every positive integer $n$, $s(n)$ denotes the sum of the digits in the decimal representation of $n$. Prove that for every integer $n \ge 5$, we have $$S(1)S(3)...S(2n-1) \ge S(2)S(4)...S(2n)$$

A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values ​​of $n$.

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system. Show that there is an integer $k$, with no $9$ in it's decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

We have a positive integer $n$, whose sum of digits is $100$ . If the sum of digits of $44n$ is $800$ then what is the sum of digits of $3n$?

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?

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$

For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.

Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?

Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively. 1) Find all values of n such that $n = P(n)$: 2) Determine all values of n such that $n = S(n) + P(n)$.

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

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?

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .