Olympiad problems

2006 MOP Homework, 3

Tags: divisor , odd , greatest , inequalities , number theory

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

2022 European Mathematical Cup, 2

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

2025 Kyiv City MO Round 2, Problem 2

Tags: divisor , number theory

A positive integer $ n $ satisfies the following conditions: [list] [*] The number $ n $ has exactly $ 60 $ divisors: $ 1 = a_1 < a_2 < \cdots < a_{60} = n $; [*] The number $ n+1 $ also has exactly $ 60 $ divisors: $ 1 = b_1 < b_2 < \cdots < b_{60} = n+1 $. [/list] Let $ k $ be the number of indices $ i $ such that $ a_i < b_i $. Find all possible values of $ k $. [i]Note: Such numbers exist, for example, the numbers $ 4388175 $ and $ 4388176 $ both have $ 60 $ divisors.[/i] [i]Proposed by Anton Trygub[/i]

2019 AMC 10, 19

Tags: divisor

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

2001 All-Russian Olympiad, 4

Tags: divisor , relatively prime , number theory

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$.

2002 Mexico National Olympiad, 5

Tags: divisor , maximum , sum , number theory

A [i]trio [/i] is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which [i]trio [/i] contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all [i]trios [/i] with the maximum sum.

2022 Bulgarian Autumn Math Competition, Problem 10.3

Tags: divisor , number theory

Are there natural number(s) $n$, such that $3^n+1$ has a divisor in the form $24k+20$

1998 Estonia National Olympiad, 1

Tags: number theory , greatest common divisor , divisor

Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: divisor , product , number theory

Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?

2022 IFYM, Sozopol, 7

Tags: divisor , number theory

Let’s note the set of all integers $n>1$ which are not divisible by a square of a prime number. We define the number $f(n)$ as the greatest amount of divisors of $n$ which could be chosen in such way so that for each two chosen $a$ and $b$, not necessarily different, the number $a^2+ab+b^2+n$ is not a square. Find all $m$ for which there exists $n$ so that $f(n)=m$.

2013 Dutch Mathematical Olympiad, 4

Tags: equation , product , positive , number theory , divisor

For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$. (a) Find all positive integers $n$ satisfying $P(n) = 15n$. (b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.

2019 239 Open Mathematical Olympiad, 2

Tags: number of divisors , divisor , number theory

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

2018 Saudi Arabia BMO TST, 3

Tags: number theory , divide , divisor

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

2021 Romanian Master of Mathematics Shortlist, N1

Tags: unit fractions , divisor , number theory

Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.

2013 Portugal MO, 4

Tags: factorial , divisor , number theory

Which is the leastest natural number $n$ such that $n!$ has, at least, $2013$ divisors?

2021 Ukraine National Mathematical Olympiad, 8

Tags: divisor , number theory

Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$. Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$. (Fedir Yudin)

2018 Dutch IMO TST, 2

Tags: divisor , number theory

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

2003 Estonia National Olympiad, 4

Tags: divisor , sum , reciprocal , number theory

Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

2003 France Team Selection Test, 3

Tags: divisor , prime , algebra , 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.

1992 ITAMO, 3

Tags: number theory , divisor , sum of divisors , factorial

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

1992 Austrian-Polish Competition, 6

Tags: divisor , functional , functional equation , algebra , number theory

A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2018 Polish Junior MO Finals, 3

Tags: number theory , coloring , divisor , combinatorics

Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible.

2015 Dutch BxMO/EGMO TST, 1

Tags: divisor , divide , number theory

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2009 Bundeswettbewerb Mathematik, 2

Tags: number theory , divisor , quadratic trinomial , trinomial , prime , prime divisor

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

2019 Thailand TST, 1

Tags: arithmetic function , number of divisors , divisor , number theory

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.