Olympiad problems

2005 iTest, 17

On the $2004$ iTest, we defined an [i]optimus [/i] prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$, which is also a prime number.) Given that you select a prime number under $100$, find the probability that is it not an optimus prime.

2018 Swedish Mathematical Competition, 4

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

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

2020 Argentina National Olympiad, 1

Tags: number theory , divide , sum of digits

For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

VMEO III 2006 Shortlist, N14

Tags: digit , sum of digits , number theory

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: $A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149}$ $T(n_1)=T(n_2)=...=T(n_{148})$ $T(m_1)=T(m_2)=...=T(m_{148})$

2013 Thailand Mathematical Olympiad, 5

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

Find a five-digit positive integer $n$ (in base $10$) such that $n^3 - 1$ is divisible by $2556$ and which minimizes the sum of digits of $n$.

1990 IMO Shortlist, 8

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}).$

2020 LIMIT Category 2, 13

Tags: sum of digits , limit

For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.

1995 Tournament Of Towns, (482) 6

Tags: arithmetic sequence , sum of digits , number theory

Does there exist an increasing arithmetic progression of (a) $11$ (b) $10000$ (c) infinitely many positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression? (A Shapovalov)

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

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 $$S(3^n) \ge S(3^{n+1}).$$

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 $6$.

2003 Cuba MO, 7

Tags: number theory , sum of digits

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

2012 May Olympiad, 2

Tags: number theory , sum of digits

We call S $(n)$ the sum of the digits of the integer $n$. For example, $S (327)=3+2+7=12$. Find the value of $$A=S(1)-S(2)+S(3)-S(4)+...+S(2011)-S(2012).$$ ($A$ has $2012$ terms).

2013 Cuba MO, 3

Tags: number theory , sum of digits

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

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 $2022$.

2004 District Olympiad, 1

Tags: number theory , sum of digits

Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $2,0,0,4.$ [hide= original wording] before finding a typo .. Find the number of positive $6$ digit integers, such that the sum of their digits is $9$, and four of its digits are $1,0,0,4.$ Posts 2 and 3 reply to this wording [/hide]

2015 Saudi Arabia IMO TST, 3

Tags: combinatorics , sum of digits , 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

2019 Bulgaria EGMO TST, 3

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

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.

2005 All-Russian Olympiad Regional Round, 8.5

Tags: number theory , sum of digits , even

It is known that the sum of the digits of the natural number $N$ is $100$, and the sum of the digits of the number $5N$ is $50$. Prove that $N$ is even.

2006 Switzerland - Final Round, 3

Tags: number theory , sum of digits , digit

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.

1996 Nordic, 1

Tags: integer , sum of digits , number theory

Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

1903 Eotvos Mathematical Competition, 1

Tags: sum of digits , number theory

Let $n = 2^{p-1} (2^p - 1)$, and let $2^p- 1$ be a prime number. Prove that the sum of all (positive) divisors of $n$ (not including $n$ itself) is exactly $n$.

2017 Taiwan TST Round 2, 1

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]

2018 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , sum of digits , divisibility

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]

2020 Costa Rica - Final Round, 1

Tags: number theory , sum of digits

Find all the $4$-digit natural numbers, written in base $10$, that are equal to the cube of the sum of its digits.