Olympiad problems

2017 Danube Mathematical Olympiad, 1

What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\in\mathbb{N}$ can take?

2018 Lusophon Mathematical Olympiad, 3

Tags: number theory , sum of digits , minimum

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.

2015 May Olympiad, 4

Tags: sum of digits , digit , number theory

We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.

2004 German National Olympiad, 3

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

Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

2017 Peru IMO TST, 14

Tags: number theory , polynomial , 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]

2022 Brazil EGMO TST, 3

Tags: number theory , divisible , sum of digits

A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?

1993 Nordic, 4

Tags: number theory , sum of digits , positive integer

Denote by $T(n)$ the sum of the digits of the decimal representation of a positive integer $n$. a) Find an integer $N$, for which $T(k \cdot N)$ is even for all $k, 1 \le k \le 1992, $ but $T(1993 \cdot N)$ is odd. b) Show that no positive integer $N$ exists such that $T(k \cdot N)$ is even for all positive integers $k$.

1992 Tournament Of Towns, (341) 3

Tags: number theory , sum of digits , divisible

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)

Oliforum Contest III 2012, 2

Tags: polynomial , sum of digits , number theory

Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

1984 Tournament Of Towns, (056) O4

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

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

2000 Switzerland Team Selection Test, 4

Tags: sum , sum of digits , number theory

Let $q(n)$ denote the sum of the digits of a natural number $n$. Determine $q(q(q(2000^{2000})))$.

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.

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2022/2023 Tournament of Towns, P5

Tags: number theory , sum of digits

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]

1989 Chile National Olympiad, 1

Tags: number theory , sum of digits , digit

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 $x,y,z,b.$

2008 Hanoi Open Mathematics Competitions, 1

Tags: number theory , divisible , sum of digits

How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?

1997 Tournament Of Towns, (524) 1

Tags: number theory , sum of digits , divisible

How many integers from $1$ to $1997$ have the sum of their digits divisible by $5$? (AI Galochkin)

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 $m$ )

2017 Morocco TST-, 6

Tags: number theory , polynomial , 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]

2016 India PRMO, 3

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

2016 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: sum of digits , sequence , divisibility , number theory

When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$, (ii) $n$ divides $m$, (iii) $n$ divides the sum of the digits of $A_m$.

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)

1993 All-Russian Olympiad Regional Round, 11.1

Tags: sum of digits , number theory

Find all natural numbers $n$ for which the sum of digits of $5^n$ equals $2^n$.

2001 Paraguay Mathematical Olympiad, 3

Tags: number theory , sum of digits , divisible

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: 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 $n$.