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

1903 Eotvos Mathematical Competition, 1
Tags: number theory , sum of digits
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 $n$.

1954 Moscow Mathematical Olympiad, 263
Tags: ratio , maximum , 3-digit , sum of digits
Define the maximal value of the ratio of a three-digit number to the sum of its digits.

1974 Czech and Slovak Olympiad III A, 3
Tags: number theory , digit , permutation , equation , diophantine equation , sum of digits
Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

1975 All Soviet Union Mathematical Olympiad, 217
Tags: number theory , polynomial , sequence , sum of digits
Given a polynomial $P(x)$ with a) natural coefficients; b) integer coefficients; Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.

2014 Thailand TSTST, 3
Tags: number theory , sum of digits
Let $s(n)$ denote the sum of digits of a positive integer $n$. Prove that $s(9^n) > 9$ for all $n\geq 3$.

2008 JBMO Shortlist, 3
Tags: sum of digits , number theory
Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

2022/2023 Tournament of Towns, P6
Tags: number theory , 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?

1997 Swedish Mathematical Competition, 5
Tags: number theory , sum of 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 $0 < k < f(n)$.

2008 Hanoi Open Mathematics Competitions, 1
Tags: divisible , sum of digits , number theory
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?

1985 Tournament Of Towns, (084) T5
Tags: algebra , combinatorics , number theory , sequence , sum of digits
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ? (S. Fomin , Leningrad)

1995 India Regional Mathematical Olympiad, 3
Tags: number theory , modular arithmetic , sum of digits
Prove that among any $18$ consecutive three digit numbers there is at least one number which is divisible by the sum of its digits.

2020 Iran Team Selection Test, 5
Tags: number theory , sum of digits
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]

2001 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: number theory , sum of digits , sequence
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 $f(1),f(2),...,f(m),...$

2011 Romania National Olympiad, 4
Tags: number theory , sum of digits
A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

2009 China Northern MO, 8
Tags: number theory , sum of digits , divide
Find the smallest positive integer $N$ satisfies : 1 . $209$│$N$ 2 . $ S (N) = 209 $ ( # Here $S(m)$ means the sum of digits of number $m$ )

2020 Costa Rica - Final Round, 1
Tags: sum of digits , number theory
Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.

2020 Durer Math Competition Finals, 4
Tags: sum of digits , number theory
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 $3n$?

2005 iTest, 17
Tags: number theory , sum of digits , prime
On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.

2019 India PRMO, 21 incorrect
Tags: sum of digits , remainder , number theory
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 $E$

2021 Malaysia IMONST 1, 9
Tags: number theory , sum of digits
Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$?

1993 Tournament Of Towns, (387) 5
Tags: number theory , sum of digits
Let $S(n)$ denote the sum of digits of $n$ (in decimal representation). Do there exist three different natural numbers $n$, $p$ and $q$ such that $$n +S(n) = p + S(p) = q + S(q)?$$ (M Gerver)

2018 Hanoi Open Mathematics Competitions, 13
Tags: number theory , sum of digits , product , digit , sum
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 $n = S(n) + P(n)$.

1992 Tournament Of Towns, (341) 3
Tags: divisible , number theory , sum of digits
Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd. (D Fomin, St Petersburg)

2010 Cuba MO, 2
Tags: number theory , sum of digits
Let $n = (p^2 +2)^2 -9(p^2 -7)$ where $p$ is a prime number. Determine the smallest value of the sum of the digits of $n$ and for what prime number $p$ is obtained.

1993 Swedish Mathematical Competition, 1
Tags: number theory , sum of digits , divisible , divide
An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.