Olympiad problems

2013 IFYM, Sozopol, 3

The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , Divisors , Integer

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

2003 Estonia National Olympiad, 4

Tags: number theory , Sum , reciprocal , Divisors

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.

1981 Austrian-Polish Competition, 7

Tags: prime divisors , Divisors , number theory

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

2004 Thailand Mathematical Olympiad, 5

Tags: Divisors , number theory

Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.

2018 Turkey Junior National Olympiad, 1

Tags: Turkey , Divisors , number theory

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

2022 European Mathematical Cup, 1

Tags: number theory , modular arithmetic , Divisibility , Divisors

Determine all positive integers $n$ for which there exist positive divisors $a$, $b$, $c$ of $n$ such that $a>b>c$ and $a^2 - b^2$, $b^2 - c^2$, $a^2 - c^2$ are also divisors of $n$.

1996 All-Russian Olympiad Regional Round, 9.5

Tags: number theory , Divisors , Sum

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

2017 China Team Selection Test, 1

Tags: number theory , Divisors , 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 $f(n) \leq n^{0.01}$.

1998 IMO, 3

Tags: number theory , Divisors , Number theoretic functions , prime factorization , IMO , IMO 1998 , IMO Shortlist

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

2020 Switzerland Team Selection Test, 4

Tags: number theory , relatively prime , Russia , Divisors , MONT

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$.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: Divisors , number theory

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

2024 AMC 10, 8

Tags: AMC , AMC 10 , 2024 AMC , 2024 AMC 10b , Divisors

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 $

1984 IMO Shortlist, 3

Tags: number theory , Divisors , equation , IMO Shortlist

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

2024 Francophone Mathematical Olympiad, 4

Tags: number theory , number theory proposed , prime numbers , Divisors

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2015 Latvia Baltic Way TST, 15

Tags: number theory , Divisors , prime divisor , prime divisors

Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.

2010 QEDMO 7th, 5

Tags: number theory , Divisors

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 $$\sum_{k\in D(n)} d(k)^3=\left( \sum_{k\in D(n)} d(k)\right) ^2.$$

2024 Belarusian National Olympiad, 10.1

Tags: number theory , Divisors

Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$ [i]M. Zorka[/i]

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , Divisors , divisor , Perfect Square

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.

2002 IMO Shortlist, 2

Tags: inequalities , number theory , Divisors , IMO , IMO 2002 , IMO Shortlist

Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , Divisors

For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$

2022 Durer Math Competition Finals, 4

Tags: number theory , Perfect Squares , Divisors

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.

2025 AIME, 7

Tags: 2025 AIME II , AMC , 2025 AMC , AIME , probability , Divisors , combinatorics

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2024 VJIMC, 4

Tags: function , Divisors , Sequences , Integers , number theory

Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.) a) Prove that $(b_n)_{n \ge 0}$ is unbounded. b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.

2012 Kyiv Mathematical Festival, 4

Tags: number theory , Divisors

Find all positive integers $a, b,c$ greater than $1$, such that $ab + 1$ is divisible by $c, bc + 1$ is divisible by $a$ and $ca + 1$ is divisible by $b$.