Olympiad problems

2018 IMO Shortlist, N1

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2018 Singapore Junior Math Olympiad, 4

Tags: factor , sum , number theory , divisor

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

2019 239 Open Mathematical Olympiad, 2

Tags: number theory , number of divisors , divisor

Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

2005 India IMO Training Camp, 2

Tags: number theory , equation , divisor

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , divisor , sum , divide , combinatorics

A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.

2024 Baltic Way, 19

Tags: construction , fraction , number theory , divisor

Does there exist a positive integer $N$ which is divisible by at least $2024$ distinct primes and whose positive divisors $1 = d_1 < d_2 < \ldots < d_k = N$ are such that the number \[ \frac{d_2}{d_1}+\frac{d_3}{d_2}+\ldots+\frac{d_k}{d_{k-1}} \] is an integer?

2020 South Africa National Olympiad, 1

Tags: number theory , divisor

Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.

2013 Irish Math Olympiad, 8

Tags: number theory , diophantine equation , divisor

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2013 Saudi Arabia IMO TST, 3

Tags: digit , divisor , number theory

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

2016 Balkan MO Shortlist, N2

Tags: divisor , maximum , number theory , odd

Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

2017 Ecuador Juniors, 5

Tags: number theory , consecutive , coprime , divisor

Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.

2020 Durer Math Competition Finals, 9

Tags: number theory , divisor

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $18$. We know that the sum of all numbers on the paper having exactly one proper divisor is $666$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a [i]proper divisor [/i]of the positive integer $n$ if $k | n$ and $1 < k < n$.

2022 All-Russian Olympiad, 1

Tags: number theory , all russia mo , divisor

We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , integer , divisor

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

2018 Regional Competition For Advanced Students, 4

Tags: number of divisors , number theory , divisor

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

2017 Regional Competition For Advanced Students, 4

Tags: number theory , sum of squares , divisibility , divisor

Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

2009 Flanders Math Olympiad, 2

Tags: number theory , divisor

A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

2015 Puerto Rico Team Selection Test, 1

Tags: number theory , divisor

A sequence of natural numbers is written according to the following rule: [i] the first two numbers are chosen and thereafter, in order to write a new number, the sum of the last numbers is calculated using the two written numbers, we find the greatest odd divisor of their sum and the sum of this greatest odd divisor plus one is the following written number. [/i]The first numbers are $25$ and $126$ (in that order), and the sequence has $2015$ numbers. Find the last number written.

2007 Indonesia TST, 2

Tags: prime divisor , number theory , divisor

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

2018 Pan-African Shortlist, N6

Tags: number theory , arithmetic mean , geometric mean , divisor

Prove that there are infinitely many integers $n$ such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers. (Recall that for $k$ positive real numbers, $a_1, a_2, \dotsc, a_k$, the arithmetic mean is $\frac{a_1 +a_2 +\dotsb +a_k}{k}$, and the geometric mean is $\sqrt[k]{a_1 a_2\dotsb a_k}$.)

2000 Czech and Slovak Match, 3

Tags: number theory , divisor , power of 2

Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.

2013 NZMOC Camp Selection Problems, 12

Tags: prime divisor , divisor , prime , number theory , inequalities

For a positive integer $n$, let $p(n)$ denote the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m-1) < p(m) < p(m + 1)$.

1998 IMO Shortlist, 6

Tags: number theory , number theoretic functions , prime factorization , divisor

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

2009 Germany Team Selection Test, 2

Tags: function , number theory , modular arithmetic , divisor , functional equation

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

2018 IFYM, Sozopol, 2

Tags: number theory , greatest common divisor , divisor

$n > 1$ is an odd number and $a_1, a_2, . . . , a_n$ are positive integers such that $gcd(a_1, a_2, . . . , a_n) = 1$. If $d = gcd (a_1^n + a_1.a_2. . . a_n, a_2^n + a_1.a_2. . . a_n, . . . , a_n^n + a_1.a_2. . . a_n) $ find all possible values of $d$.