Olympiad problems

1984 Tournament Of Towns, (071) T5

Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

2017 India PRMO, 1

Tags: number theory , sum of digits , divide , 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 $3$?

2021 China Team Selection Test, 3

Tags: sum of digits , decimal representation , number theory

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 $n$.

2020 Iran MO (3rd Round), 3

Tags: number theory , function , sum of digits

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.

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 $n\geq 3$.

2022 IFYM, Sozopol, 4

Tags: digit , sum of digits , number theory

A natural number $x$ is written on the board. In one move, we can take the number on the board and between any two of its digits in its decimal notation we can we put a sign $+$, or we may not put it, then we calculate the obtained result and we write it on the board in place of $x$. For example, from the number $819$. we can get $18$ by $8 + 1 + 9$, $90$ by $81 + 9$, and $27$ by $8 + 19$. Prove that no matter what $x$ is, we can reach a single digit number with at most $4$ moves.

2018 India PRMO, 28

Tags: sum of digits , combinatorics

Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.

2000 Belarus Team Selection Test, 4.3

Tags: algebra , modular arithmetic , arithmetic sequence , divisibility , sum of digits

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 $M$.

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)

2015 Latvia Baltic Way TST, 14

Tags: sum of digits , number theory

Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?

2020 Durer Math Competition Finals, 3

Tags: number theory , sum of digits

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?

1956 Moscow Mathematical Olympiad, 321

Tags: number theory , 2-digit , sum of digits

Find all two-digit numbers $x$ the sum of whose digits is the same as that of $2x$, $3x$, ... , $9x$.

2009 Grand Duchy of Lithuania, 1

Tags: number theory , multiple , sum of digits

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.

2024 Czech-Polish-Slovak Junior Match, 5

Tags: sum of digits , number theory

For a positive integer $n$, let $S(n)$ be the sum of its decimal digits. Determine the smallest positive integer $n$ for which $4 \cdot S(n)=3 \cdot S(2n)$.

1999 Bundeswettbewerb Mathematik, 2

Tags: number theory , sum of digits , digit , inequalities

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 $Q(3^k) \ge Q(3^{k+1})$.

2014 Lithuania Team Selection Test, 4

Tags: last digit , number theory , sum of digits

(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

1989 Mexico National Olympiad, 3

Tags: number theory , sum of digits

Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

1975 All Soviet Union Mathematical Olympiad, 217

Tags: number theory , sequence , polynomial , 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.

2019 Pan-African Shortlist, C2

Tags: number theory , combinatorics , sum of digits

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]

2017 Saudi Arabia IMO TST, 1

Tags: algebra , polynomial , number theory , 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 $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.

2011 Abels Math Contest (Norwegian MO), 1

Tags: number theory , sum of digits , sequence

Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$, that is, $n = 1234567891011...40214022$. a. Show that $n$ is divisible by $3$. b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$

1985 Tournament Of Towns, (084) T5

Tags: sequence , algebra , combinatorics , sum of digits , number theory

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)

2017 Romanian Master of Mathematics Shortlist, N1

Tags: sum of digits , number theory

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: $$S(2^{24^{2017}}k)=S(k)$$

1997 Dutch Mathematical Olympiad, 1

Tags: number theory , sum of digits , product

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

2000 Belarus Team Selection Test, 5.2

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 $k$.