Olympiad problems

2020 AIME Problems, 9

Tags: divisor , probability

Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m.$

2008 Korea Junior Math Olympiad, 6

Tags: sum of powers , sum , divisible , number theory , 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 $8$.

2014 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , sets of integers , divisor

A pair (a,b) of positive integers is [i]Rioplatense [/i]if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is [i]Rioplatense[/i].

2001 Estonia National Olympiad, 2

Tags: number theory , least common multiple , divide , divisor

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

2024 Bulgaria MO Regional Round, 11.3

Tags: extremal , divisor , number theory

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: perfect square , number theory , divisor

Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

2010 QEDMO 7th, 5

Tags: divisor , number theory

For a natural number $n$, let $D (n)$ be the set of (positive integers) divisors of $n$. Furthermore let $d (n)$ be the number of divisors of $n,$ that is, the cardinality of $D (n)$. For each such $n$, prove the equality $$\sum_{k\in D(n)} d(k)^3=\left( \sum_{k\in D(n)} d(k)\right) ^2.$$

2019 Bosnia and Herzegovina EGMO TST, 2

Tags: divisor , number theory

Let $1 = d_1 < d_2 < ...< d_k = n$ be all natural divisors of the natural number $n$. Find all possible values of the number $k$ if $n=d_2d_3 + d_2d_5+d_3d_5$.

2016 Argentina National Olympiad, 5

Tags: divisor , algebra , prime divisor , number theory

Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.

2018 Israel Olympic Revenge, 1

Tags: prime number , divisibility , number theory , divisor

Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

2019 Tournament Of Towns, 1

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

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)

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: divisor , reciprocal sum , sum , irreducible , number theory

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2015 Puerto Rico Team Selection Test, 1

Tags: divisor , number theory

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.

2009 Germany Team Selection Test, 2

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

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]

2009 Switzerland - Final Round, 10

Tags: divisor , number theory

Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.

2017 China Team Selection Test, 1

Tags: divisor , number theory , modular arithmetic

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

1983 Poland - Second Round, 4

Tags: divisor , sum , number theory

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

2022 Durer Math Competition Finals, 4

Tags: perfect square , divisor , number theory

Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

2015 Postal Coaching, Problem 4

Tags: number theory , divisor

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

2018 Turkey Junior National Olympiad, 1

Tags: number theory , divisor

Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

2015 Czech-Polish-Slovak Junior Match, 5

Tags: divisor , number theory

Determine all natural numbers$ n> 1$ with the property: For each divisor $d> 1$ of number $n$, then $d - 1$ is a divisor of $n - 1$.

2015 Kyiv Math Festival, P3

Tags: divisor , number theory

Is it true that every positive integer greater than 30 is a sum of 4 positive integers such that each two of them have a common divisor greater than 1?

2016 Greece JBMO TST, 3

Tags: greatest common divisor , number theory , divisor

Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$

2016 Danube Mathematical Olympiad, 2

Tags: prime number , divisor , number theory

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

2011 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: divisor , number theory , algebra

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.