Olympiad problems

2023 Greece Junior Math Olympiad, 4

Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.

2013 Irish Math Olympiad, 8

Tags: number theory , Diophantine equation , Divisors

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2024 Brazil Cono Sur TST, 3

Tags: number theory , Divisors , Sigma function , tau function , ceiling function

Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]

Mathematical Minds 2024, P1

Tags: number theory , Divisors

Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

2009 Bundeswettbewerb Mathematik, 2

Tags: number theory , quadratic trinomial , trinomial , prime divisors , prime , Divisors

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.

2017 Bosnia and Herzegovina EGMO TST, 3

Tags: Sequence , number theory , Divisors

For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$

2018 Estonia Team Selection Test, 8

Tags: Divisors , 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 Dürer Math Competition (First Round), P3

Tags: number theory , Divisors

a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Show that if the product of all the positive divisors of two natural numbers are equal, then the two numbers must be equal.

2005 Germany Team Selection Test, 1

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 $n$.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , reciprocal sum , Sum , Divisors , Irreducible

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2016 AMC 10, 22

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? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

2002 IMO, 4

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 $n^2$.

2011 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: Divisors , number theory , algebra

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$. Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.

2018 Saudi Arabia IMO TST, 1

Tags: prime divisors , Divisors , primes , theory

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

2013 Portugal MO, 4

Tags: number theory , factorial , Divisors

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

2004 Estonia Team Selection Test, 5

Tags: number theory , LCM , power of 2 , Divisors

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: Divisors , number theory , number of divisors

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

2021 Ukraine National Mathematical Olympiad, 8

Tags: number theory , Divisors

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)

2011 Cuba MO, 3

Tags: number theory , Divisors

Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$ the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

2023 ISL, N1

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 $1 \leq i \leq k - 2$.

2016 Switzerland Team Selection Test, Problem 7

Tags: number theory , Divisors

Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$

2025 Kyiv City MO Round 2, Problem 2

Tags: number theory , Divisors

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]

2001 Switzerland Team Selection Test, 4

Tags: number theory , Sum , Divisors

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 Peru MO (ONEM), 3

Tags: number theory , consecutive , Divisors

We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.

2014 Contests, 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 $d_4+d_5=d_6+7$.