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

2012 Singapore Senior Math Olympiad, 2
Tags: square , sum of digits , number theory
Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

2000 Belarus Team Selection Test, 5.2
Tags: sum of digits , decimal representation , number theory
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 $k$.

2016 India PRMO, 3
Tags: digit , number theory , sum of digits , algebra
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.

2012 May Olympiad, 2
Tags: sum of digits , number theory
We call S $(n)$ the sum of the digits of the integer $n$. For example, $S (327)=3+2+7=12$. Find the value of $$A=S(1)-S(2)+S(3)-S(4)+...+S(2011)-S(2012).$$ ($A$ has $2012$ terms).

2004 District Olympiad, 1
Tags: number theory , sum of digits
Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$ [hide= original wording] before finding a typo .. Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$ Posts 2 and 3 reply to this wording [/hide]

2020 Argentina National Olympiad, 1
Tags: sum of digits , divide , number theory
For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

2000 Austria Beginners' Competition, 3
Tags: product of digits , digit , sum of digits , number theory
A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?

2010 Belarus Team Selection Test, 8.1
Tags: root , sum of digits , number theory
The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$. a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots. b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots. (I. Voronovich)

2017 Peru IMO TST, 14
Tags: sum of digits , number theory , polynomial , functional equation
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]

2001 Cuba MO, 3
Tags: number theory , sum of digits
Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.

2024 Czech-Polish-Slovak Junior Match, 5
Tags: number theory , sum of digits
For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n)=3 \cdot S(2n)$.

2013 Thailand Mathematical Olympiad, 5
Tags: number theory , sum of digits , digit , divide , divisible
Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

2017 Ukraine Team Selection Test, 7
Tags: functional equation , number theory , sum of digits , polynomial
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]

1982 All Soviet Union Mathematical Olympiad, 329
Tags: divisible , sum of digits , exponential , number theory
a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?

2016 India Regional Mathematical Olympiad, 3
Tags: number theory , inequalities , sum of digits
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.

2007 Cuba MO, 8
Tags: periodic sequence , periodic , number theory , sum of digits
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.

2016 IMO Shortlist, N1
Tags: functional equation , polynomial , sum of digits , number theory
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]

2020 LIMIT Category 2, 13
Tags: sum of digits , limit
For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.

2019 Mediterranean Mathematics Olympiad, 3
Tags: number theory , decimal representation , sum of digits
Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$. (Proposed by Gerhard Woeginger, Austria)

2021 China Team Selection Test, 3
Tags: number theory , sum of digits , decimal representation
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 $n$.

2012 Tournament of Towns, 7
Tags: sum of digits , number theory
Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?

2019 Pan-African Shortlist, C2
Tags: sum of digits , combinatorics , number theory
On the board, we write the integers $1, 2, 3, \dots, 2019$. At each minute, we pick two numbers on the board $a$ and $b$, delete them, and write down the number $s(a + b)$ instead, where $s(n)$ denotes the sum of the digits of the integer $n$. Let $N$ be the last number on the board at the end. [list=a] [*] Is it possible to get $N = 19$? [*] Is it possible to get $N = 15$? [/list]

2005 All-Russian Olympiad Regional Round, 8.5
Tags: even , number theory , sum of digits
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.

2011 May Olympiad, 1
Tags: sum of digits , number theory
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 $x$).

1999 IMO Shortlist, 5
Tags: number theory , decimal representation , sum of digits
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 $k$.