Olympiad problems

2012 Singapore Senior Math Olympiad, 2

Determine all positive integers $n$ such that $n$ equals the square of the sum of the digits of $n$.

2000 Austria Beginners' Competition, 3

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

A two-digit number is [i]nice [/i] if it is both a multiple of the product of its digits and a multiple of the sum of its digits. How many numbers satisfy this property? What is the ratio of the number to the sum of digits for each of the nice numbers?

2016 Hanoi Open Mathematics Competitions, 2

Tags: sum of digits , number theory

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.

2011 Romania National Olympiad, 4

Tags: number theory , sum of digits

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

1997 Belarusian National Olympiad, 1

Tags: number theory , sum of digits

$$Problem1:$$ A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?

2005 Paraguay Mathematical Olympiad, 1

Tags: number theory , sum of digits

With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written?

2008 JBMO Shortlist, 3

Tags: sum of digits , number theory

Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$.

Russian TST 2017, P1

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]

2017 Estonia Team Selection Test, 11

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]

2015 Romania National Olympiad, 4

Tags: number theory , sum of digits

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

1995 India Regional Mathematical Olympiad, 3

Tags: number theory , modular arithmetic , sum of digits

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

2001 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: sum of digits , number theory , sequence

For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let $f(m)=m+S(m)$. Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$

2007 Abels Math Contest (Norwegian MO) Final, 1

Tags: number theory , sum of digits

We consider the sum of the digits of a positive integer. For example, the sum of the digits of $2007$ is equal to $9$, since $2 + 0 + 0 + 7 = 9$. (a) How many integers $n$, where $0 < n < 100 000$, have an even sum of digits? (b) How many integers $n$, where $0 < n < 100 000$, have a sum of digits that is less than or equal to $22$?

2017 Lusophon Mathematical Olympiad, 3

Tags: number theory , sum of digits

Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.

2007 Argentina National Olympiad, 5

Tags: number theory , sum of digits

We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?

2022 Swedish Mathematical Competition, 3

Tags: number theory , sum of digits

Let $n$ be a positive integer divisible by $39$. What is the smallest possible sum of digits that $n$ can have (in base $10$)?

2003 May Olympiad, 3

Tags: number theory , multiple , sum of digits

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

2017 Estonia Team Selection Test, 11

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]

2021 Malaysia IMONST 1, 9

Tags: number theory , sum of digits

Find the sum of (decimal) digits of the number $(10^{2021} + 2021)^2$?

2001 Cuba MO, 3

Tags: number theory , sum of digits

Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.

1974 Czech and Slovak Olympiad III A, 3

Tags: number theory , digit , sum of digits , permutation , equation , diophantine equation

Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]

2007 Cuba MO, 8

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

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.

2012 Tournament of Towns, 7

Tags: sum of digits , number theory

Peter and Paul play the following game. First, Peter chooses some positive integer $a$ with the sum of its digits equal to $2012$. Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is $2012$. On each of his moves Paul chooses a positive integer $x$ and Peter tells him the sum of the digits of $|x - a|$. What is the minimal number of moves in which Paul can determine Peter’s number for sure?

2017 Ukraine Team Selection Test, 7

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]

1982 All Soviet Union Mathematical Olympiad, 329

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

a) Let $m$ and $n$ be natural numbers. For some nonnegative integers $k_1, k_2, ... , k_n$ the number $$2^{k_1}+2^{k_2}+...+2^{k_n}$$ is divisible by $(2^m-1)$. Prove that $n \ge m$. b) Can you find a number, divisible by $111...1$ ($m$ times "$1$"), that has the sum of its digits less than $m$?