Olympiad problems

1998 Belarus Team Selection Test, 1

Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$). Prove that $(S(n))^3 <n^4$.

2020 European Mathematical Cup, 2

Tags: number theory , fibonacci , divisor , sequence

A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$. Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]

2019 Tournament Of Towns, 1

Tags: inequalities , number theory , prime , number of divisors , prime factorization , factor , divisor

Let us call the number of factors in the prime decomposition of an integer $n > 1$ the complexity of $n$. For example, [i]complexity [/i] of numbers $4$ and $6$ is equal to $2$. Find all $n$ such that all integers between $n$ and $2n$ have complexity a) not greater than the complexity of $n$. b) less than the complexity of $n$. (Boris Frenkin)

2022 European Mathematical Cup, 2

Tags: number theory , greatest common divisor , divisibility , divisor

We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$. a) Are there infinitely many lovely numbers? b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?

2003 France Team Selection Test, 3

Tags: algebra , divisor , prime , number theory

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

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)

2024 Baltic Way, 16

Tags: perfect power , number theory , divisor

Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.

2020 Switzerland Team Selection Test, 4

Tags: number theory , relatively prime , divisor

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divisor

For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$

2025 AIME, 7

Tags: probability , combinatorics , divisor

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2001 Switzerland Team Selection Test, 4

Tags: number theory , sum , divisor

For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.

2015 Balkan MO Shortlist, N3

Tags: prime divisor , integer , number theory , divide , divisor

Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

2014 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: number theory , number of divisors , divisor

El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?

2017 International Zhautykov Olympiad, 2

Tags: number theory , divisor , prime

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2002 Junior Balkan Team Selection Tests - Romania, 1

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

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

2016 Flanders Math Olympiad, 2

Tags: power , number theory , divide , divisor

Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.

2018 Saudi Arabia IMO TST, 1

Tags: prime divisor , theory , divisor , prime

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

2023 Regional Olympiad of Mexico West, 1

Tags: number theory , divisor

For every positive integer $n$ we take the greatest divisor $d$ of $n$ such that $d\leq \sqrt{n}$ and we define $a_n=\frac{n}{d}-d$. Prove that in the sequence $a_1,a_2,a_3,...$, any non negative integer $k$ its in the sequence infinitely many times.

1992 Austrian-Polish Competition, 1

Tags: sum , positive , number theory , product , even , divisor

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: natural number , number theory , algebra , polynomial , divisor

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

2017 South East Mathematical Olympiad, 4

Tags: number theory , calculate , divisor

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$. Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.

1997 Tuymaada Olympiad, 6

Tags: number theory , consecutive , divisor

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

2004 IMO Shortlist, 1

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 $n$.

2004 Estonia Team Selection Test, 5

Tags: number theory , lcm , power of 2 , divisor

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2019 Romania National Olympiad, 1

Tags: inequalities , number theory , divisor

Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$ where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$. Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.