Olympiad problems

2005 Germany Team Selection Test, 1

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 $n$.

2018 Switzerland - Final Round, 7

Tags: number theory , consecutive , Divisors

Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$. Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.

Russian TST 2019, P1

Tags: number theory , Divisors

A positive integer $n{}$ is called [i]discontinuous[/i] if for all its natural divisors $1 = d_0 < d_1 <\cdots<d_k$, written out in ascending order, there exists $1 \leqslant i \leqslant k$ such that $d_i > d_{i-1}+\cdots+d_1+d_0+1$. Prove that there are infinitely many positive integers $n{}$ such that $n,n+1,\ldots,n+2019$ are all discontinuous.

2013 Regional Competition For Advanced Students, 1

Tags: Divisors , probability , number theory

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

2017 Regional Competition For Advanced Students, 4

Tags: number theory , Sum of Squares , Divisors , Divisibility

Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

1995 All-Russian Olympiad Regional Round, 9.5

Tags: number theory , Divisors

Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).

2018 Singapore Junior Math Olympiad, 4

Tags: Sum , factors , Divisors , number theory

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

2024 Iberoamerican, 1

Tags: number theory , Divisors , GCD and LCM

For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \] and determine all pairs of positive integers $(a,b)$ where we have equality case.

2012 QEDMO 11th, 12

Tags: number theory , Divisors , divides , divisible

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

2005 India IMO Training Camp, 2

Tags: number theory , Divisors , equation , IMO Shortlist

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 $n$.

2016 Federal Competition For Advanced Students, P1, 4

Tags: Austria , Divisors , ratio , number theory

Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)

2016 Danube Mathematical Olympiad, 2

Tags: number theory , prime numbers , Divisors , romania

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

2016 Greece JBMO TST, 3

Tags: number theory , greatest common divisor , Divisors

Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$

2018 Poland - Second Round, 2

Tags: number theory , combinatorics , Divisors , Poland , inequalities

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

2023 IMO, 1

Tags: IMO , IMO 2023 , number theory , Divisors

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

2017 International Zhautykov Olympiad, 2

Tags: number theory , primes , Divisors

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2002 Switzerland Team Selection Test, 3

Tags: Divisors , number theory , Sum

Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.

1998 Belarus Team Selection Test, 3

Tags: number theory , Divisors

Let $1=d_1<d_2<d_3<...<d_k=n$ be all different divisors of positive integer $n$ written in ascending order. Determine all $n$ such that $$d_7^2+d_{10}^2=(n/d_{22})^2.$$

2022 Saudi Arabia BMO + EGMO TST, 2.2

Tags: number theory , Divisors

Find all positive integers $n$ that have precisely $\sqrt{n + 1}$ natural divisors.

2021 Romanian Master of Mathematics Shortlist, N1

Tags: RMM Shortlist , number theory , unit fractions , Divisors

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 $N$.

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , number theory , Divisors , Difference

Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.

2006 Portugal MO, 5

Tags: number theory , Divisors

Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.

2007 Indonesia TST, 2

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.

2017 China National Olympiad, 5

Tags: number theory , Divisors

Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.

2018 Israel Olympic Revenge, 1

Tags: prime numbers , Divisibility , number theory , Divisors

Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$