Olympiad problems

2010 Belarus Team Selection Test, 8.1

The function $f : N \to N$ is defined by $f(n) = n + S(n)$, where $S(n)$ is the sum of digits in the decimal representation of positive integer $n$. a) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has no natural roots. b) Prove that there are infinitely many numbers $a \in N$ for which the equation $f(x) = a$ has at least two distinct natural roots. (I. Voronovich)

1990 IMO Longlists, 23

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

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}).$

1997 Swedish Mathematical Competition, 5

Tags: number theory , sum of digits

Let $s(m)$ denote the sum of (decimal) digits of a positive integer $m$. Prove that for every integer $n > 1$ not equal to $10$ there is a unique integer $f(n) \ge 2$ such that $s(k)+s(f(n)-k) = n$ for all integers $k$ with $0 < k < f(n)$.

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 $2002$.

2023 Romania National Olympiad, 3

Tags: number theory , sum of digits

Determine all natural numbers $m$ and $n$ such that \[ n \cdot (n + 1) = 3^m + s(n) + 1182, \] where $s(n)$ represents the sum of the digits of the natural number $n$.

2021 Girls in Mathematics Tournament, 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?

1954 Moscow Mathematical Olympiad, 263

Tags: maximum , ratio , 3-digit , sum of digits

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

2016 IMO Shortlist, N1

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 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 $a,b<10^{10}$.

2011 Kyiv Mathematical Festival, 2

Tags: number theory , sum of digits

Is it possible to represent number $2011... 2011$, where number $2011$ is written $20112011$ times, as a product of some number and sum of its digits?

Kvant 2024, M2783

Tags: number theory , sum of digits

The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a] [*] the square of this number; [*]the fourth power of this number, [/list] given that $k\geqslant 4.$ [i]From the folklore[/i]

2023 Romania Team Selection Test, P1

Tags: number theory , sum of digits

Let $m$ and $n$ be positive integers, where $m < 2^n.$ Determine the smallest possible number of not necessarily pairwise distinct powers of two that add up to $m\cdot(2^n- 1).$ [i]The Problem Selection Committee[/i]

2015 India PRMO, 6

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

$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

2015 Caucasus Mathematical Olympiad, 5

Tags: number theory , combinatorics , sum of digits , palindrome

Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?

2001 Estonia National Olympiad, 2

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

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

1980 All Soviet Union Mathematical Olympiad, 294

Tags: sum of digits , number theory

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 $n$.

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 $S(n) \equiv n \mod p$.

2002 Greece JBMO TST, 2

Tags: number theory , digit , 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.

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.

2020 Iran Team Selection Test, 5

Tags: number theory , sum of digits

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]

2004 Chile National Olympiad, 4

Tags: sum of digits , digit , number theory

Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).

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

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]

2011 May Olympiad, 1

Tags: number theory , sum of digits

Find a positive integer $x$ such that the sum of the digits of $x$ is greater than $2011$ times the sum of the digits of the number $3x$ ($3$ times $x$).

2013 Junior Balkan Team Selection Tests - Romania, 2

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

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 $81$.