Olympiad problems

2016 Dutch IMO TST, 3

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

2018 Lusophon Mathematical Olympiad, 3

Tags: minimum , sum of digits , number theory

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 $S (n) -S (n + a) = 2018$.

2017 Danube Mathematical Olympiad, 1

Tags: number theory , sum of digits

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

2013 Junior Balkan Team Selection Tests - Romania, 2

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

Call the number $\overline{a_1a_2... a_m}$ ($a_1 \ne 0,a_m \ne 0$) the reverse of the number $\overline{a_m...a_2a_1}$. Prove that the sum between a number $n$ and its reverse is a multiple of $81$ if and only if the sum of the digits of $n$ is a multiple of $81$.

VMEO III 2006 Shortlist, N14

Tags: digit , number theory , sum of digits

For any natural number $n = \overline{a_i...a_2a_1}$, consider the number $$T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i.$$ Let's find the smallest positive integer $A$ such that is sum of the natural numbers $n_1,n_2,...,n_{148}$ as well as of $m_1,m_2,...,m_{149}$ and matches the pattern: $A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$ $T(n_1)=T(n_2)=...=T(n_{148})$ $T(m_1)=T(m_2)=...=T(m_{148})$

2023 Romania National Olympiad, 2

Tags: algebra , sum of digits

We say that a natural number is called special if all of its digits are non-zero and any two adjacent digits in its decimal representation are consecutive (not necessarily in ascending order). a) Determine the largest special number $m$ whose sum of digits is equal to $2023$. b) Determine the smallest special number $n$ whose sum of digits is equal to $2022$.

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?

1992 Czech And Slovak Olympiad IIIA, 3

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

Let $S(n)$ denote the sum of digits of $n \in N$. Find all $n$ such that $S(n) = S(2n) = S(3n) =... = S(n^2)$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.3

Tags: sum of digits , number theory

Find any two consecutive natural numbers, each of which is divisible by the square of the sum of its digits.

1987 Austrian-Polish Competition, 7

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

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

2001 All-Russian Olympiad Regional Round, 8.6

Tags: sum of digits , number theory

We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?

2020 Iran Team Selection Test, 5

Tags: sum of digits , number theory

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]

2006 Switzerland - Final Round, 3

Tags: digit , sum of digits , number theory

Calculate the sum of digit of the number $$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$ where the number of nines doubles in each factor.

2017 China Northern MO, 7

Tags: number theory , sum of digits

Let $S(n)$ denote the sum of the digits of the base-10 representation of an natural number $n$. For example. $S(2017) = 2+0+1+7 = 10$. Prove that for all primes $p$, there exists infinitely many $n$ which satisfy $S(n) \equiv n \mod p$.

Russian TST 2017, P4

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

1984 Tournament Of Towns, (056) O4

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

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?

2003 Singapore Team Selection Test, 1

Tags: number theory , sum of digits

Determine whether there exists a positive integer $n$ such that the sum of the digits of $n^2$ is $2002$.

2018 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , divisibility , sum of digits

For each positive integer $n$ let $s(n)$ denote the sum of the decimal digits of $n$. Find all pairs of positive integers $(a, b)$ with $a > b$ which simultaneously satisfy the following two conditions $$a \mid b + s(a)$$ $$b \mid a + s(b)$$ [i]Proposed by Victor Domínguez[/i]

2017 Taiwan TST Round 2, 1

Tags: number theory , functional equation , 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]

2003 Cuba MO, 7

Tags: sum of digits , number theory

Let S(n) be the sum of the digits of the positive integer $n$. Determine $$S(S(S(2003^{2003}))).$$

2008 Bulgarian Autumn Math Competition, Problem 8.3

Tags: number theory , prime number , sum of digits

Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.

1999 Portugal MO, 4

Tags: sum of digits , number theory

Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?

2022 IFYM, Sozopol, 4

Tags: number theory , digit , sum of digits

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.

1990 IMO Shortlist, 8

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}).$

2016 Hanoi Open Mathematics Competitions, 2

Tags: number theory , sum of digits

The number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$ is (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.