Olympiad problems

2022/2023 Tournament of Towns, P6

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?

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 $n=2S(n)^2$.

2018 Brazil National Olympiad, 6

Tags: number theory , sum of digits

Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression $S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.

1999 IMO Shortlist, 6

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 $M$.

2016 Dutch IMO TST, 3

Tags: number theory , sum of digits , digit

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 $s(n) > s(2n)$.

2021 Czech-Polish-Slovak Junior Match, 6

Tags: number theory , sum of digits

Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.

1987 Austrian-Polish Competition, 7

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

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 $Q$.

1993 Swedish Mathematical Competition, 1

Tags: sum of digits , divisible , divide , number theory

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 $9$.

Kvant 2022, M2724

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]