Olympiad problems

2019 China Team Selection Test, 2

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 $$\sum_{d|n,d<n,d \in S} d = n$$

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: number theory , Product , Products , Divisors

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 ?

1997 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , Divisors

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.

1986 IMO Longlists, 58

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

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

2019 AMC 10, 9

Tags: AMC 10 , 2019 AMC 10A , 2019 AMC , AMC , Divisors , Divisibility

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers? $\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

2014 EGMO, 3

Tags: quadratics , number theory , Divisors , Combinatorial Number Theory , EGMO , EGMO 2014

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

2013 IMO Shortlist, N7

Tags: algebra , number theory , Divisors , IMO Shortlist

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if \[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good. Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.

2020 German National Olympiad, 4

Tags: number theory , number theory proposed , Divisibility , Divisors

Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.

2022 Bulgarian Autumn Math Competition, Problem 10.3

Tags: number theory , Divisors

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

2001 All-Russian Olympiad Regional Round, 9.6

Tags: number theory , last digits , Digits , Divisors

Is there such a natural number that the product of all its natural divisors (including $1$ and the number itself) ends exactly in $2001$ zeros?

2014 IFYM, Sozopol, 2

Tags: number theory , Divisors

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

2020 Final Mathematical Cup, 1

Tags: number theory , Divisors , Perfect Squares

Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.

1992 Austrian-Polish Competition, 6

Tags: functional , functional equation , Divisors , 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?

2016 AMC 12/AHSME, 18

Tags: 2016 AMC 10A , Divisors , 2016 AMC 12A , AMC 10A , AMC 12A , AMC 12 , number theory

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 Alborz Mathematical Olympiad, P1

Tags: number theory , Divisors , complete residue system , AlborzMO

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari

2019 Romania National Olympiad, 1

Tags: inequalities , number theory , Divisors

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$.

2019 AMC 10, 19

Tags: 2019 AMC 12B , 2019 AMC , AMC , AMC 12 , Divisors , 2019 AMC 10B , AMC 10

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

1983 IMO Shortlist, 2

Tags: number theory , Divisors , sum of divisors , Inequality , IMO Shortlist

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2009 Argentina National Olympiad, 2

Tags: number theory , Divisors

A positive integer $n$ is [i]acceptable [/i] if the sum of the squares of its proper divisors is equal to $2n+4$ (a divisor of $n$ is [i]proper [/i] if it is different from $1$ and of $n$ ). Find all acceptable numbers less than $10000$,

2020 Dürer Math Competition (First Round), P1

Tags: number theory , Divisors

a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?

BIMO 2021, 1

Tags: number theory , Divisors , Perfect Numbers

Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$. Prove that every natural number $n$ has a multiple that is good.

2022 China Team Selection Test, 5

Tags: number theory , Divisors , binomial coefficients

Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent: (a) For any positive divisor $m$ of $n$, \[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \] (b) For any positive divisor $k$ of $n$, \[ k ~\Big|~ \sum_{d|k} f(d). \]

2012 Tournament of Towns, 4

Tags: number theory , prime , Divisors

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

2006 Cuba MO, 5

Tags: number theory , Divisors , primes

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.

2008 Thailand Mathematical Olympiad, 2

Tags: Divisors , number theory

Find all positive integers $N$ with the following properties: (i) $N$ has at least two distinct prime factors, and (ii) if $d_1 < d_2 < d_3 < d_4$ are the four smallest divisors of $N$ then $N =d_1^2 + d_2 ^2+ d_3 ^2+ d_4^2$