Olympiad problems

1980 All Soviet Union Mathematical Olympiad, 284

All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?

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 .

2007 Switzerland - Final Round, 8

Tags: combinatorics , number theory , divisible

Let $M\subset \{1, 2, 3, . . . , 2007\}$ a set with the following property: Among every three numbers one can always choose two from $M$ such that one is divisible by the other. How many numbers can $M$ contain at most?

2008 Korea Junior Math Olympiad, 6

Tags: number theory , sum of powers , sum , divisible , divisor

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

2018 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , divide , divisible

For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

1979 Chisinau City MO, 174

Tags: number theory , odd , divide , divisible

Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.

2009 Hanoi Open Mathematics Competitions, 11

Tags: combinatorics , number theory , divisible

Let $A = \{1,2,..., 100\}$ and $B$ is a subset of $A$ having $48$ elements. Show that $B$ has two distint elements $x$ and $y$ whose sum is divisible by $11$.

2018 Czech-Polish-Slovak Junior Match, 4

Tags: digit , number theory , divisible

Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

2017 Hanoi Open Mathematics Competitions, 14

Tags: divisible , number theory

Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$ , where $m, n \in N$. a) Is $P$ divisible by $24$? b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$?

2021 New Zealand MO, 3

Tags: number theory , divisible

In a sequence of numbers, a term is called [i]golden [/i] if it is divisible by the term immediately before it. What is the maximum possible number of golden terms in a permutation of $1, 2, 3, . . . , 2021$?

1996 Estonia National Olympiad, 4

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

Prove that, for each odd integer $n \ge 5$, the number $1^n+2^n+...+15^n$ is divisible by $480$.

1951 Kurschak Competition, 2

Tags: number theory , divide , divisible

For which $m > 1$ is $(m -1)!$ divisible by $m$?

2012 India Regional Mathematical Olympiad, 2

Tags: power , number theory , divide , divisible

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

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 $n$.

2025 Romania EGMO TST, P2

Tags: divisible , number theory , inequalities , subset

Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$

2005 Thailand Mathematical Olympiad, 19

Tags: algebra , polynomial , divisible

Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$, the remainder when $P(x)$ is divided by $x - k$ is equal to $k$. Find the value of $P(4) + P(0)$.

2015 JBMO Shortlist, NT2

Tags: number theory , divisible , repunit

A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted as $\underbrace{{11\cdots1}}_{n}$ . Prove that: α) the repunit $\underbrace{{11\cdots1}}_{n}$is divisible by $37$ if and only if $n$ is divisible by $3$ b) there exists a positive integer $k$ such that the repunit $\underbrace{{11\cdots1}}_{n}$ is divisible by $41$ if $n$ is divisible by $k$

1982 Tournament Of Towns, (015) 1

Tags: number theory , divisible , divisor

Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. (M Levin)

1987 Tournament Of Towns, (159) 3

Tags: number theory , divisible , divide

Prove that there are infinitely many pairs of natural numbers $a$ and $b$ such that $a^2 + 1$ is divisible by $b$ and $b^2 + 1$ is divisible by $a$ .

2014 Junior Balkan Team Selection Tests - Moldova, 4

Tags: number theory , divide , divisible

A set $A$ contains $956$ natural numbers between $1$ and $2014$, inclusive. Prove that in the set $A$ there are two numbers $a$ and $b$ such that $a + b$ is divided by $19$.

2017 Auckland Mathematical Olympiad, 3

Tags: number theory , divisible , digit , divide

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

1993 All-Russian Olympiad Regional Round, 11.2

Tags: number theory , divide , divisible

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

2019 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible

Let $a_0$ be an arbitrary positive integer. Let $(a_n)$ be infinite sequence of positive integers such that for every positive integer $n$, the term $a_n$ is the smallest positive integer such that $a_0 + a_1 +... + a_n$ is divisible by $n$. Prove that there exist $N$ such that $a_{n+1} = a_n$ for all $n \ge N$

2019 Saudi Arabia Pre-TST + Training Tests, 5.1

Tags: number theory , prime , divide , divisible

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

2022 APMO, 1

Tags: number theory , divisible , multiple

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.