Olympiad problems

2008 Korea Junior Math Olympiad, 6

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 Pan-African Shortlist, N6

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

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 $\sqrt[k]{a_1 a_2\dotsb a_k}$.)

2001 Estonia National Olympiad, 4

Tags: number theory , harmonic , Divisors

We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.

2016 EGMO, 6

Tags: number theory , EGMO , Divisors , EGMO 2016

Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.

2014 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: number theory , number of divisors , Divisors

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?

2024 Mexico National Olympiad, 2

Tags: number theory , Mexico , Divisors

Determine all pairs $(a, b)$ of integers that satisfy both: 1. $5 \leq b < a$ 2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order. [b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.

2015 Azerbaijan National Olympiad, 4

Tags: Divisors , prime divisor , number theory

Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2022 Greece National Olympiad, 2

Tags: number theory , Divisors

Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$ for all possible values of $n$, as well as for which positive integers $n$ this minimum value is attained.

1998 Belarus Team Selection Test, 1

Tags: inequalities , number theory , Sum , Divisors

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

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: number theory , Divisors , Natural Number , algebra , polynomial

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.

2012 Tournament of Towns, 2

Tags: number theory , Divisors

The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?

2017 Iberoamerican, 5

Tags: Divisors , Iberoamerican , combinatorics

Given a positive integer $n$, all of its positive integer divisors are written on a board. Two players $A$ and $B$ play the following game: Each turn, each player colors one of these divisors either red or blue. They may choose whichever color they wish, but they may only color numbers that have not been colored before. The game ends once every number has been colored. $A$ wins if the product of all of the red numbers is a perfect square, or if no number has been colored red, $B$ wins otherwise. If $A$ goes first, determine who has a winning strategy for each $n$.

2021 Bangladeshi National Mathematical Olympiad, 10

Tags: combinatorics , counting , number theory , Divisors

A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.

2022 China Team Selection Test, 6

Tags: number theory , Divisibility , Divisors

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

2010 Saudi Arabia Pre-TST, 3.2

Tags: Perfect Square , Divisors , number theory

Prove that among any nine divisors of $30^{2010}$ there are two whose product is a perfect square.

2023 Israel TST, P2

Tags: TST , number theory , Divisors

For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example, \[A(10)=1+2+5+10=18\] \[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\] Find all positive integers $n$ for which $A(n)$ divides $B(n)$.

2016 All-Russian Olympiad, 3

Tags: number theory , Divisors , greatest common divisor

Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

2020 Malaysia IMONST 1, 17

Tags: Divisors , number theory

Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

2020 South Africa National Olympiad, 1

Tags: number theory , Divisors

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

2022 Durer Math Competition Finals, 16

Tags: number theory , Divisors

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

1986 IMO Shortlist, 6

Tags: number theory , Divisors , prime factorization , IMO Shortlist

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

1998 Switzerland Team Selection Test, 6

Tags: primes , Divisors , number theory

Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.

2019 China Team Selection Test, 2

Tags: Divisors , number theory

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$

2019 Saudi Arabia IMO TST, 2

Tags: number theory , Divisors

Find all pair of integers $(m,n)$ and $m \ge n$ such that there exist a positive integer $s$ and a) Product of all divisor of $sm, sn$ are equal. b) Number of divisors of $sm,sn$ are equal.

2019 AIME Problems, 9

Tags: AMC , AIME , AIME II , 2019 AIME II , 2019 AMC , Divisors

Call a positive integer $n$ $k$[i]-pretty[/i] if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$[i]-pretty[/i]. Let $S$ be the sum of positive integers less than $2019$ that are $20$[i]-pretty[/i]. Find $\tfrac{S}{20}$.