Olympiad problems

1980 All Soviet Union Mathematical Olympiad, 294

Let us denote with $S(n)$ the sum of all the digits of $n$. a) Is there such an $n$ that $n+S(n)=1980$? b) Prove that at least one of two arbitrary successive natural numbers is representable as $n + S(n)$ for some third number $n$.

1997 Dutch Mathematical Olympiad, 1

Tags: sum of digits , product , number theory

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 $f (n) = 19091997$.

2013 India PRMO, 6

Tags: number theory , sum of digits

Let $S(M)$ denote the sum of the digits of a positive integer $M$ written in base $10$. Let $N$ be the smallest positive integer such that $S(N) = 2013$. What is the value of $S(5N + 2013)$?

2015 Saudi Arabia IMO TST, 3

Tags: sum of digits , combinatorics , number theory

Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_1, I_2$ of $S$ of the same length, we have • The sum of digits of $I_1$ differs from the sum of digits of $I_2$ by at most $1$, • If $I_1$ begins on the left end of S then the sum of digits of $I_1$ is not greater than the sum of digits of $I_2$, • If $I_2$ ends on the right end of S then the sum of digits of $I_2$ is not less than the sum of digits of $I_1$. Lê Anh Vinh

2023 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , perfect square , sum of digits

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

2023 Turkey Olympic Revenge, 3

Tags: polynomial , sum of digits , number theory

Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$ for all $x,y\in \mathbb{N}$. Note: $s(x)$ denotes the sum of digits of $x$. [i]Proposed by Şevket Onur YILMAZ[/i]

2013 Cuba MO, 3

Tags: number theory , sum of digits

Find all the natural numbers that are $300$ times the sum of its digits.

1983 Tournament Of Towns, (035) O4

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

The natural numbers $M$ and $K$ are represented by different permutations of the same digits. Prove that (a) The sum of the digits of $2M$ equals the sum of the digits of $2K$. (b) The sum of the digits of $M/2$ equals the sum of the digits of $K/2$ ($M, K$ both even). (c) The sum of the digits of $5M$ equals the sum of the digits of $5 K$. (AD Lisitskiy)

1998 IMO Shortlist, 7

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

Prove that for each positive integer $n$, there exists a positive integer with the following properties: It has exactly $n$ digits. None of the digits is 0. It is divisible by the sum of its digits.

1997 All-Russian Olympiad Regional Round, 11.3

Tags: number theory , sum of digits

Let us denote by $S(m)$ the sum of the digits of the natural number $m$. Prove that there are infinitely many positive integers $n$ such that $$S(3^n) \ge S(3^{n+1}).$$

2014 Lithuania Team Selection Test, 4

Tags: last digit , sum of digits , number theory

(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?

2021 Israel TST, 1

Tags: sum of digits , number theory

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

2021 Kyiv City MO Round 1, 9.2

Tags: sum of digits , number theory

Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$? [i]Proposed by Oleksii Masalitin[/i]

2009 Argentina National Olympiad, 6

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

A sequence $a_0,a_1,a_2,...,a_n,...$ is such that $a_0=1$ and, for each $n\ge 0$ , $a_{n+1}=m \cdot a_n$ , where $m$ is an integer between $2$ and $9$ inclusive. Also, every integer between $2$ and $9$ has even been used at least once to get $a_{n+1} $ from $a_n$ . Let $Sn$ the sum of the digits of $a_n$ , $n=0,1,2,...$ . Prove that $S_n \ge S_{n+1}$ for infinite values of $n$.

2017 Saudi Arabia IMO TST, 1

Tags: sum of digits , polynomial , number theory , algebra

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 $S(P(n)) = P(S(n))$.

2016 Hanoi Open Mathematics Competitions, 8

Tags: sum , perfect cube , number theory , sum of digits

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

2003 May Olympiad, 3

Tags: multiple , sum of digits , number theory

Find the smallest positive integer that ends in $56$, is a multiple of $56$, and has the sum of its digits equal to $56$.

2019 Bulgaria EGMO TST, 3

Tags: sum of digits , upper bound on sequence , ratio , number theory

In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \] as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.

2009 Grand Duchy of Lithuania, 1

Tags: multiple , number theory , 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.

2011 JBMO Shortlist, 5

Tags: number theory , sum of digits

Find the least positive integer such that the sum of its digits is $2011$ and the product of its digits is a power of $6$.

1993 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , sum of digits

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

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 $q(q(q(2000^{2000})))$.

2002 Greece JBMO TST, 2

Tags: digit , number theory , sum of digits

Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number. Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.

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 $p, q, r$.

Kvant 2023, M2763

Tags: number theory , sum of digits , modular arithmetic

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]