This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 159

2006 Junior Balkan Team Selection Tests - Romania, 3
Tags: divisor , sum of digits , number theory
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 $n$.

1997 Belarusian National Olympiad, 1
Tags: sum of digits , number theory
$$Problem1:$$ A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?

1997 Tournament Of Towns, (524) 1
Tags: sum of digits , divisible , number theory
How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

2007 Argentina National Olympiad, 5
Tags: sum of digits , number theory
We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?

1999 IMO Shortlist, 6
Tags: divisibility , arithmetic sequence , algebra , sum of digits , modular arithmetic
Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

2004 Chile National Olympiad, 4
Tags: sum of digits , digit , number theory
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

2001 Estonia National Olympiad, 2
Tags: digit , remainder , number theory , sum of digits
Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

2022 Swedish Mathematical Competition, 3
Tags: sum of digits , number theory
Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

2023 Romania Team Selection Test, P1
Tags: number theory , sum of digits
Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$ [i]The Problem Selection Committee[/i]

2017 Romanian Master of Mathematics Shortlist, N1
Tags: number theory , sum of digits
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: $$S(2^{24^{2017}}k)=S(k)$$

2017 Estonia Team Selection Test, 11
Tags: polynomial , number theory , functional equation , sum of 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]

2017 India PRMO, 1
Tags: sum of digits , divide , number theory , divisible
How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

1987 Polish MO Finals, 3
Tags: sequence , sum of digits , algebra , number theory , polynomial
$w(x)$ 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 $p_1, p_2, p_3, ...$ .

2018 Brazil National Olympiad, 6
Tags: number theory , sum of digits
Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression $S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.

1996 Nordic, 1
Tags: integer , number theory , sum of digits
Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

1990 IMO Longlists, 23
Tags: digit , calculate , decimal representation , sum of digits , number theory
For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

1989 Chile National Olympiad, 1
Tags: digit , number theory , sum of digits
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 $x,y,z,b.$

2001 Paraguay Mathematical Olympiad, 3
Tags: number theory , divisible , sum of digits
Find a $10$-digit number, in which no digit is zero, that is divisible by the sum of their digits.

2021 China Team Selection Test, 3
Tags: decimal representation , number theory , sum of digits
Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following: There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements, $$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$ where $S(n)$ denotes sum of digits of decimal representation of $n$.

2011 Kyiv Mathematical Festival, 2
Tags: sum of digits , number theory
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?

2018 Swedish Mathematical Competition, 4
Tags: consecutive , divisible , number theory , sum of digits
Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

2020 Iran MO (3rd Round), 3
Tags: sum of digits , function , number theory
Find all functions $f$ from positive integers to themselves, such that the followings hold. $1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$. $2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.

2021 Israel TST, 1
Tags: sum of digits , number theory
A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

2012 CentroAmerican, 1
Tags: digit sum , sum of digits , number theory
Find all positive integers that are equal to $700$ times the sum of its digits.

Kvant 2022, M2724
Tags: sum of digits , number theory
In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]