Olympiad problems

2022 Bundeswettbewerb Mathematik, 4

For each positive integer $k$ let $a_k$ be the largest divisor of $k$ which is not divisible by $3$. Let $s_n=a_1+a_2+\dots+a_n$. Show that: (a) The number $s_n$ is divisible by $3$ iff the number of ones in the ternary expansion of $n$ is divisible by $3$. (b) There are infinitely many $n$ for which $s_n$ is divisible by $3^3$.

2021 Dutch Mathematical Olympiad, 5

Tags: divisor , number theory , prime

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

2024 AMC 10, 8

Tags: divisor

Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? $ \textbf{(A) }0 \qquad \textbf{(B) }2 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

1982 Bundeswettbewerb Mathematik, 1

Tags: number theory , divisor , sum

Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

2018 India PRMO, 1

Tags: divisor , number theory , prime

A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times that in the second. The sum of the page numbers on the first pages of the three volumes is $1709$. If $n$ is the last page number, what is the largest prime factor of $n$?

1996 All-Russian Olympiad Regional Round, 9.5

Tags: number theory , divisor , sum

Find all natural numbers that have exactly six divisors whose sum is $3500$.

2018 Estonia Team Selection Test, 8

Tags: divisor , number theory

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2020 Durer Math Competition Finals, 5

Tags: divisor , number theory

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $30$. We know that the sum of all numbers on the paper having exactly one proper divisor is $2397$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a proper divisor of the positive integer $n$ if $k | n$ and $1 < k < n$.

2014 Czech-Polish-Slovak Junior Match, 5

Tags: game , number theory , divide , divisor

There is the number $1$ on the board at the beginning. If the number $a$ is written on the board, then we can also write a natural number $b$ such that $a + b + 1$ is a divisor of $a^2 + b^2 + 1$. Can any positive integer appear on the board after a certain time? Justify your answer.

2018 IFYM, Sozopol, 2

Tags: greatest common divisor , number theory , 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 $d$.

2016 AMC 10, 22

Tags: number theory , divisor

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

2024 Mexico National Olympiad, 2

Tags: number theory , divisor

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 $x < d < y$.

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: number theory , divisor

Find the three-digit number that has the greatest number of different divisors.

1980 Dutch Mathematical Olympiad, 2

Tags: number theory , divisor

Find the product of all divisors of $1980^n$, $n \ge 1$.

2005 India IMO Training Camp, 2

Tags: number theory , divisor , equation

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 $n$.

2020 Malaysia IMONST 1, 17

Tags: number theory , divisor

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 $6n$.

2013 Regional Competition For Advanced Students, 1

Tags: number theory , probability , divisor

For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?

2002 BAMO, 4

Tags: inequalities , number theory , divisor

For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$. Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds. For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$ .

1989 Bundeswettbewerb Mathematik, 4

Tags: egyptian fractions , divisor , number theory

Let $n$ be an odd positive integer. Show that the equation $$ \frac{4}{n} =\frac{1}{x} + \frac{1}{y}$$ has a solution in the positive integers if and only if $n$ has a divisor of the form $4k+3$.

2016 EGMO, 6

Tags: number theory , divisor

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.

2001 Switzerland Team Selection Test, 4

Tags: number theory , divisor , sum

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 $n \ge 2$.

2007 Thailand Mathematical Olympiad, 16

Tags: number theory , divisor

What is the smallest positive integer with $24$ positive divisors?

2019 Saudi Arabia IMO TST, 2

Tags: number theory , divisor

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.

2015 European Mathematical Cup, 3

Tags: number theory , divisor

Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

2000 Portugal MO, 3

Tags: divisor , number theory , divide

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.